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Geometric methods for the physics of magnetisedplasmas
Eric SonnendruckerMax-Planck-Institut fur Plasmaphysik
andZentrum Mathematik, TU Munchen
Lecture notesWintersemester 2019/2020
January 31, 2020
Contents
1 Introduction 21.1 The physics background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 What are geometric methods? . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Variational principles 62.1 Motion of a particle in a electrostatic field . . . . . . . . . . . . . . . . . . . 6
2.1.1 Newton’s equations of motion . . . . . . . . . . . . . . . . . . . . . . 62.1.2 Coordinate system and motion . . . . . . . . . . . . . . . . . . . . . 62.1.3 Derivation of the Euler-Lagrange equations . . . . . . . . . . . . . . . 72.1.4 Hamilton’s canonical equations . . . . . . . . . . . . . . . . . . . . . 10
2.2 Motion of a particle in an electromagnetic field . . . . . . . . . . . . . . . . . 132.2.1 Newton’s equations of motion . . . . . . . . . . . . . . . . . . . . . . 132.2.2 Action principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2.3 Hamilton’s equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2.4 The phase-space Lagrangian . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 The Vlasov-Maxwell equations . . . . . . . . . . . . . . . . . . . . . . . . . . 172.3.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.3.2 Action principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3 Numerical methods for hamiltonian systems 213.1 Splitting methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2 Numerical integration of the motion of a charged particle in a static electro-
magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2.1 Hamiltonian splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2.2 Divergence free splitting . . . . . . . . . . . . . . . . . . . . . . . . . 253.2.3 Discrete gradient methods . . . . . . . . . . . . . . . . . . . . . . . . 27
4 Some useful notions from differential geometry 294.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.2 Cotangent plane and covectors . . . . . . . . . . . . . . . . . . . . . . . . . . 324.3 p-forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.4 Riemannian metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.5 Exterior derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.6 Interior product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.7 Orientation and twisted forms . . . . . . . . . . . . . . . . . . . . . . . . . . 374.8 Scalar product of p-forms and Hodge ? operator . . . . . . . . . . . . . . . . 384.9 Pullback of differential forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
1
4.10 Maxwell’s equations with differential forms . . . . . . . . . . . . . . . . . . . 40
5 Finite Element Exterior Calculus (FEEC) 435.1 The Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 435.2 The de Rham complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.3 Discrete differential forms based on B-splines . . . . . . . . . . . . . . . . . 45
5.3.1 B-Splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.3.2 Discrete differential forms in 1D . . . . . . . . . . . . . . . . . . . . . 475.3.3 Discrete differential forms in 3D . . . . . . . . . . . . . . . . . . . . . 51
5.4 Discretisation of Maxwell’s equations with spline differential forms . . . . . . 555.5 Classical Finite Element approximation of Maxwell’s equations . . . . . . . . 585.6 Derivation of PIC model from discrete action principle . . . . . . . . . . . . 59
2
Abstract
Many models in plasma physics have been shown to possess a Hamiltonian structure and canbe derived from an action principle. This is true for the equations of motion of a particlein a given electromagnetic field, which is a finite dimensional Hamiltonian system, for whichsymplectic numerical integrators have been derived with a large success and a well devel-oped theory. Many other, more complex models for plasma physics, disregarding dissipativeeffects, also fit into an infinite dimensional non canonical hamiltonian geometric structure.This geometric structure provides the basis for the conservation of some fundamental physicsinvariants like energy, momentum and some Casimir invariants like Gauss’ law and div B=0.Therefore preserving it in asymptotic models, like for example the Gyrokinetic model forstrongly magnetised plasmas, or numerical approximations can be very helpful. New numer-ical methods, like mimetic Finite Differences and Finite Element Exterior Calculus basedon the discretisation of objects coming from differential geometry allow in a natural wayto preserve the geometric structure of the continuous equations. Schemes derived on theseconcepts allow on the one hand to rederive very good well-known schemes that have previ-ously been found in an ad hoc way, like the Yee scheme for Maxwell’s equations or chargeconserving Particle In Cell methods. These concepts will be introduced and applied to someclassical models from plasma physics: Maxwell’s equations, a cold plasma model and theVlasov-Maxwell equations.
Chapter 1
Introduction
1.1 The physics background
The context of this lecture is about analytical and numerical modeling of plasmas, which areglobally neutral gases of charged particles, in particular those where there is a non negligible,and even sometimes very strong, external or self-consistent magnetic field. As we will see,this adds additional challenges compared to the purely electrostatic case. Examples of suchplasmas are typically found in astrophysics and in magnetic fusion experiments.
In this lecture we will be considering a few models illustrating the different aspects wewould like to emphasise. We will restrict ourselves for the sake of simplicity to non relativisticproblems, which are mostly relevant for the physics problems we want to address. Firstthe equations of motion of a charged particle of charge q and mass m in a given staticelectromagnetic field:
dx
dt= v, (1.1)
dv
dt=
q
m(E + v ×B). (1.2)
The other main model considered on its own, will be the Maxwell equations with givensources:
− 1
c2
∂E
∂t+∇×B = µ0J, (Ampere’s law) (1.3)
∂B
∂t+∇× E = 0, (Faraday’s law) (1.4)
∇ · E =ρ
ε0, (Gauss’s law) (1.5)
∇ ·B = 0, (magnetic Gauss’s law) (1.6)
in the international unit system, ε0, µ0, c being respectively the permittivity and the perme-ability of free space and the speed of light, verifying ε0µ0c
2 = 1.Our main model for the self-consistent evolution of a plasma will be the Vlasov-Maxwell
model. We are neglecting the collisions throughout this lecture as they need to be modelledseparately, using in geometric models the metriplectic Ansatz [16], where the full modelis composed of a conservative, symplectic, part, and a dissipative part (the collisions). The
2
Vlasov equation describing the evolution of the phase space density of each species of particlesfs reads
∂fs∂t
+ v · ∇xfs +qsms
(E + v ×B) · ∇vfs = 0, (1.7)
and is coupled with Maxwell’s equations
− 1
c2
∂E
∂t+∇× B = µ0 J,
∂B
∂t+∇× E = 0,
∇ · E =ρ
ε0
,
∇ · B = 0.
The source terms for Maxwell’s equation, the charge density ρ(x, t) and the current densityJ(x, t) can be expressed from the distribution functions of the different species of particlesfs(x,v, t) using the relations
ρ(x, t) =∑s
qs
∫fs(x,v, t) dv,
J(x, t) =∑s
qs
∫fs(x,v, t)v dv.
We will see that when solving the Vlasov-Maxwell equations using a geometric discretisa-tion of the fields and a particle description of the distribution function (GEMPIC), we obtaina semi-discrete system, time remaining continuous, which has a non canonical Hamiltonianstructure.
The Vlasov-Maxwell equations, when the needed collisions are added, are considered toyield a very accurate description of the evolution of a plasma for all problems of interests.But in some cases, simpler models are also useful as they provide a faster time to solutionin a computer code. Often fluid models, obtained by taking a few velocity moments of theVlasov equations, provide a good description.
The following cold plasma model, used in particular for modelling waves in plasmas, canbe also shown to have a non canonical Hamiltonian structure, and can be solved using thetechniques that will be introduced in this lecture
− 1
c2
∂E
∂t+∇×B = µ0J, (1.8)
∂B
∂t+∇× E = 0, (1.9)
∂J
∂t= (ε0ω
2p)E− J× ωc, (1.10)
where ωp(x) and ωc(x) are the given space dependent plasma frequency and cyclotron fre-quency that characterise the plasma.
The aim of the lecture is first to derive geometric numerical approximation of these models,starting with the ordinary differential equations for the motion of charged particles that willenable us to introduce geometric methods. Then we will introduce the tools of differential
3
geometry that will be needed to properly describe our partial differential equations at thecontinuous level, so that a natural geometric discretisation in space can be performed leadingus to a a very large semi-discrete system that has a non canonical Hamiltonian structure andthat can then be discretised using the classical geometric methods presented in the first part.
We will derive the gyrokinetic model at the end of the lecture. Modern gyrokinetic theoryis an asymptotic reduction of the full kinetic Vlasov-Maxwell model in a strong and slowlyvarying external magnetic field that is valid for studying low frequency phenomena thatare well below the gyro-frequency. This model will be an example of geometric asymptoticreduction, where all the approximations are made on the Lagrangian so that the resultingmodels keeps the non canonical hamiltonian structure and has exact invariants, which areclose to the original invariants of the Vlasov-Maxwell system in the considered regime.
1.2 What are geometric methods?
Classical numerical analysis is based on the fundamental notions of consistency and stability.First one tries to find a numerical approximation of a mathematical operator, which is closeenough to this operator and converges to it when the discretisation parameter goes to zero.This yields consistency. And then one needs to make sure that the sensitivity to the datais not too large, which enables to prove that the approximate solution stays close enough tothe exact solution, when the discretisation parameter is small enough and for short enoughtimes, for time dependent problems. This is related to stability. We then get error estimatesof the form ‖u(t)− uh(t)‖ ≤ C(T )hp for an order p method.
However, if the original system of differential equations has a specific structure involvingsome symmetries or invariants, for example conservation of total energy or of volume elementsor other conservation laws, classical numerical tools do not guarantee anything about theseinvariants. This is where geometric methods come into play. Their aim is to constructapproximations that exactly preserve some invariants, which are in general different from butclose to the physical invariants. And these invariants are preserved exactly in the numericalscheme, i.e. up to round-off errors in actual simulations. These properties help in gettingmuch better long time simulations and are also a key property for the correct behaviourof strongly non linear simulations. For example in long time simulations in gravitationproblems or the motion of trapped particles in a tokamak plasma (see Figure 1.1, standardnumerical methods like Runge-Kutta do not conserve energy: with explicit Runge-Kuttamethods the orbit becomes larger than the actual orbit for explicit method and smaller forimplicit methods that are dissipative. Of course higher order methods enable to keep closerto the right orbit for a longer time. On the other hand energy preserving or symplecticintegrator stay close to the exact orbit for all time, even though the phase error can becomelarge especially for low order methods.
An example where local conservation properties play an essential role is nonlinear con-servation laws where it is well-known that only locally conservative methods can accuratelycapture shock formation and get the correct shock velocity. For Partial Differential Equations(PDEs), geometric methods are based on having an exact expression of Stokes theorem onall the parts of the grid: e.g. in 3D for any edge C = [A,B] of the grid∫
C
∇φ · d` =
∫∂C
φ = φ(B)− φ(A),
4
14 I. Examples and Numerical Experiments
Table 2.2. Data for the outer solar system
planet mass initial position initial velocity
−3.5023653 0.00565429Jupiter m1 = 0.000954786104043 −3.8169847 −0.00412490
−1.5507963 −0.00190589
9.0755314 0.00168318Saturn m2 = 0.000285583733151 −3.0458353 0.00483525
−1.6483708 0.00192462
8.3101420 0.00354178Uranus m3 = 0.0000437273164546 −16.2901086 0.00137102
−7.2521278 0.00055029
11.4707666 0.00288930Neptune m4 = 0.0000517759138449 −25.7294829 0.00114527
−10.8169456 0.00039677
−15.5387357 0.00276725Pluto m5 = 1/ (1.3 · 108) −25.2225594 −0.00170702
−3.1902382 −0.00136504
J SU
N
P
explicit Euler, h = 10
J SU
N
P
implicit Euler, h = 10
J SU
N
P
symplectic Euler, h = 100
J SU
N
P
Stormer–Verlet, h = 200
Fig. 2.4. Solutions of the outer solar system
To this system we apply the explicit and implicit Euler methods with step sizeh = 10, the symplectic Euler and the Stormer–Verlet method with much largerstep sizes h = 100 and h = 200, repectively, all over a time period of 200 000days. The numerical solution (see Fig. 2.4) behaves similarly to that for the Keplerproblem. With the explicit Euler method the planets have increasing energy, theyspiral outwards, Jupiter approaches Saturn which leaves the plane of the two-bodymotion. With the implicit Euler method the planets (first Jupiter and then Saturn)
For a bracket of the form (51) to be a good Poisson bracketit must have the following properties for all functions f,g, h:
• antisymmetry
ff ; gg ¼ "fg; fg () Jab ¼ "Jba: (52)
• Jacobi identity
ff ; fg; hggþ cyc $ 0() JadJbc;d þ cyc $ 0; (53)
where cyc means cyclic permutation over fgh in the firstexpression and over abc in the second.
This noncanonical generalization of Hamiltonian mechanicsis reasonable because of an old theorem due to Darboux,which states that if det J 6¼ 0 then there exists a coordinatechange that (at least locally) brings J into the canonical formJc of (26). Recalling that J transforms as a rank 2 contravar-iant tensor, this canonizing transformation !z $ z would satisfy
Jl! @!za
@zl
@!zb
@z!¼ Jab
c : (54)
However, the more interesting case is the one studied bySophus Lie where det J ¼ 0. This case is degenerate andgives rise to Casimir invariants (Lie’s distinguished func-tions), which are constants of motion for any possibleHamiltonian that satisfies
f ;Cf g ¼ 0 8 f () Jab @C
@zb¼ 0 8 a: (55)
Because of the degeneracy, there is no coordinate transfor-mation to canonical form; however, a theorem known to Lie(see e.g., Refs. 69 and 70) which we call the Lie-Darbouxtheorem states that there is a transformation to the followingdegenerate canonical form:
Jdc ¼0N IN 0
"IN 0N 0
0 0 0M"2N
0
B@
1
CA: (56)
Instigated in a major way by the noncanonical Poissonbrackets for plasma models, manifolds with the addition ofdegenerate Poisson bracket structure, known as Poissonmanifolds, have now been widely studied (see e.g., Ref. 71).The local structure of a Poisson manifold is depicted inFig. 5, where it is seen that the phase space is foliated by thelevel sets of the Casimir invariants. For an M-dimensionalsystem, there exist M " 2N Casimir invariants, and an orbitthat initially lies on such a surface defined by the level setsof the initial Casimir invariants remains there. These surfa-ces, called symplectic leaves, have dimension 2N and thephase space is generically foliated by them.
Lie-Poisson brackets are a special form of noncanonicalPoisson brackets that typically appear in matter models interms of an Eulerian variable description. For finite-dimensional Lie-Poisson Hamiltonian systems, the Poissonmatrix J is linear in the dynamical variable and has the formJab ¼ cab
c zc, where the numbers cabc are the structure con-
stants of some Lie algebra.Noncanonical Hamiltonian field theories have Poisson
brackets of the form
FIG. 4. Variational symplectic integra-tor vs. RK4 for producing an accuratebanana orbit. (Courtesy of Qin, seeRef. 60.) (left) Banana orbit using stan-dard RK4 with exact orbit and (right)that obtained using a variational sym-plectic method. Orbits were obtainedfor ITER parameters with the integra-tion time being 104 banana periods.Since the ITER burn time is morethan 106 banana periods, numericalfidelity over very long times isrequired. Modified with permissionfrom Qin et al., Phys. Plasmas 16,042510 (2009). Copyright 2009 AIPPublishing LLC.
FIG. 5. Depiction of a Poisson manifold foliated by symplectic leaves ofconstant Casimir invariants.
055502-9 P. J. Morrison Phys. Plasmas 24, 055502 (2017)
c©Hairer, Lubich, Wanner, ”Geometric numerical integration” c©Hong Qin et al., Physics of Plasmas 2016
Figure 1.1: Orbits of planets in solar system (left) and banana orbits in Tokamaks (right).
on any face of the grid ∫S
∇× A · dS =
∫∂S
A · d`,
and on any volume of the grid ∫V
∇ · B dV =
∫∂V
B · dS.
These relation can be discretised exactly by using adequate discrete objects, which are notonly point values like in classical numerical methods, but also edge integrals, face integralsand volume integrals. Then approximations will be needed to properly relate objects ofthis kind in the physics models. The methods that have these features are called geometricbecause they have a geometric interpretation, independent of any coordinate system. Themathematical tools for describing them are those of differential geometry.
The classical language of vector calculus gives the same name to geometrically differentobjects and hence is not precise enough to automatically derive an appropriate geometricdiscretisation. Indeed in a 3 dimensional space a scalar field can either be a 0-form ora 3-form and a vector field can either be a 1-form or a 2-form. For this reason, someformalism from differential geometry is necessary to be able to systematically find the correctgeometric discretisation. However, we will try to make this lecture as accessible as possible topractitioners and only introduce the concepts and tools that are needed to find the geometricdiscrete representation of a field, without introducing more abstractions than necessary.
5
Chapter 2
Variational principles
2.1 Motion of a particle in a electrostatic field
2.1.1 Newton’s equations of motion
Consider a given electrostatic field E = −∇φ. The equations of motion of a particle of chargeq and mass m can be derived from Newton’s law:
dx
dt= v (2.1)
dv
dt= − q
m∇φ. (2.2)
These equations depend on the coordinate system, even though the motion of a particlebetween two points can be seen geometrically as a path (a curve) that is a geometric object,that is the same independently of the coordinate system used to represent it. Is there also away to obtain a geometric description of the motion, which is independent of the coordinatesystem being used?
2.1.2 Coordinate system and motion
Consider U ⊂ Rn an n-dimensional space endowed with an origin O and a, possibly positiondependent, coordinate system (e1, . . . , en). Then any point M in U is characterised by the
vector r =−−→OM and its coordinates α = (α1, . . . , αn)>, such that
r =n∑i=1
αiei.
We use the ( )> notation to indicate that α is a column vector.
Examples:
1. Cartesian coordinates in R2: ex = (1, 0)>, ey = (0, 1)>. Then r = xe1 + ye2 so that itscoordinates are α1 = x, and α2 = y.
2. Polar coordinates in R2: er = (cos θ, sin θ)>, eθ = (− sin θ, cos θ)>. Then r = re1 andits coordinates are α1 = r, α2 = 0.
6
A motion in U can be viewed as curve parametrised by time, also called a path:
γ = r(t) : t0 ≤ t ≤ t1.
The curve is the geometric object and the path its parametrisation. A same curve cancorrespond to several different parametrisations.
The velocity vector and the acceleration vector of the motion at the point r(t) are definedby
r =dr
dt(t), r(t) =
d2r
dt2.
In cartesian coordinates the components of the velocity vector are simply (x, y) as thebasis vectors are constant, but in curvilinear coordinates the basis vector also needs to bederived. For example in polar coordinates
r(t) =dr
dter + r(t)
d
dter = r(t) er + r(t)
(−θ(t) sin θ(t)
θ(t) cos θ(t)
)= r(t) er + r(t)θ(t) eθ,
so that the components of the velocity vector are (r, rθ).
Notations: The vector containing the derivatives of a scalar function F with respect tothe variables qi, the gradient, has two classical notations in the literature:
∇qF =∂F
∂q=
∂F∂q1...∂F∂qn
.
We will use both in this lecture. ∇q (or ∂∂q
) can be thought of as a column vector with
components ∂∂q1, . . . , ∂
∂qn. For two vectors F,G, we denote by FG their tensor product which
is the matrix with components (FiGj)i,j. So for a vector field
F(q) =
F1(q1, . . . , qn)...
Fn(q1, . . . , qn)
.
∇qF is considered to be the tensor product of the vector ∇q and the vector F. This writesexplicitly
∇qF =∂F
∂q=
∂F1
∂q1. . . ∂Fn
∂q1...
...∂F1
∂qn. . . ∂Fn
∂qn
= (DF)>,
where (DF) is the Jacobian matrix of F.
2.1.3 Derivation of the Euler-Lagrange equations
We define the Lagrangian L = K − U of a mechanical system as being the difference of thekinetic energy of the particle K = 1
2m|r|2 and the potential energy U(r). Expressing the
Lagrangian in cartesian coordinates, we find
L(x, x) =1
2m|x|2 − qφ(x). (2.3)
7
Let γ be any path describing a curve starting at a point r0 and ending at a point r1
parametrised by x(t), t0 ≤ t ≤ t1 with x(t0) = r0 and x(t1) = r1. We then define theaction integral as A(γ) =
∫γL. For practical calculation, we define the action integral in
some coordinate system
A[q] =
∫ t1
t0
L(q(t), q(t), t) dt. (2.4)
Hamilton’s least action principle states that the motion of a mechanical system coincides withextremal paths of the action integral. In order to find the extremal paths, we shall considersmall variations of the action integral along an arbitrary direction h(t) at each point, withh(t) bounded, and define the directional derivative of a functional F : Rn → R, for q,h ∈ Rn
we define
δF (q; h) = h · ∇qF (q) = h · ∂F∂q
(q) = limε→0
F [q + εh]− F [q]
ε=
d
dε
∣∣∣∣ε=0
F [q + εh].
Let ε be a small parameter. Then, performing a Taylor expansion in ε of the Lagrangian onthe displaced path
L(q(t) + εh(t), q(t) + εh(t), t) = L(q(t), q(t), t) + εh · ∇qL(q(t), q(t), t)
+ εdh
dt· ∇qL(q(t), q(t), t) +O(ε2)
Note that∇q is the gradient with respect to the variables in q which is considered independentfrom q. Then we get
A[x + εh]−A[x]
ε=
1
ε
∫ t1
t0
[L(q(t) + εh(t), q(t) + εh(t), t)− L(q(t), q(t), t)
]dt, (2.5)
=
∫ t1
t0
y ·[∇qL(q(t), q(t), t)− d
dt∇qL(q(t), q(t), t)
](2.6)
+ [h(t) · ∇qL(q(t), q(t), t)]t1t0 +O(ε). (2.7)
integrating the second term by parts. Now using that the boundary term vanishes as thepath extremities are fixed, so that h(t0) = h(t1) = 0 and taking the limit when ε → 0, wefind that the extrema of the action integral are achieved when
d
dt∇qL(q(t), q(t), t) = ∇qL(q(t), q(t), t).
These equations are called the Euler-Lagrange equations of the action principle that alsoread, using the alernative notation for the gradients:
d
dt
∂L
∂q(q(t), q(t), t) =
∂L
∂q(q(t), q(t), t). (2.8)
We observe that this derivation of the Euler-Lagrange equations which are the equations ofmotion is purely geometric and consists in finding the path that minimises the variations andis true for any coordinate system q, provided of course that the Lagrangian is properly definedwith respect to these coordinates. The choice of the coordinate system, or parametrisation,
8
appears in the definition of the Lagrangian, but the form of the Euler-Lagrange equations(2.8) is the same for any coordinate system.
Definition 1 The generalised momentum is defined from the Lagrangien by
p(t) =∂L
∂q(q(t), q(t), t).
The Euler-Lagrange equations can then also be written equivalently
dp
dt=∂L
∂q(q(t), q(t), t). (2.9)
Example 2: In cartesian coordinates, for the Lagrangian (2.3), L(x, x) = 12m|x|2 − qφ(x),
where q = x, we get the Euler-Lagrange equations
d(mx)
dt= −q∇φ.
Then setting v = x, we recover (2.1)–(2.2). Moreover the generalised momentum is thephysical momentum
p =∂L
∂x= mv.
Example 2: Consider now a parametrisation in polar coordinates (r, θ). The basis vectorsare then
er =
(cos θsin θ
), eθ =
(− sin θcos θ
).
Then a point on a curve will be of the form r(t) = r(t) er(t) so that r = r er + rθ eθ and thekinetic energy is 1
2m|r|2 = 1
2m(r2 + r2θ2). Assuming that we have a central force field, i.e.
that the potential only depends on r, the Lagrangian in polar coordinates writes, denotingby q = (r, θ)> and q = (r, θ)>
L(q, q) = L(r, θ, r, θ) =1
2m(r2 + r2θ2)− qφ(r). (2.10)
The generalised momentum is in this case
∂L
∂q=
(mr
mr2θ
),
and the Euler-Lagrange equations
d
dt
(mr
mr2θ
)=∂L
∂q=
(mrθ2 − q dφ(r)
dr
0
).
Remark 1 We observe here a general feature of the Euler-Lagrange equations. If a coordi-nate does not appear in the Lagrangian (here θ), then the corresponding component of thegeneralised momentum ∂L
∂θis a constant of motion. Such a coordinate is called cyclic.
9
2.1.4 Hamilton’s canonical equations
Definition 2 Given a Lagrangian L(q, q, t) such that the equation defining the canonicalmomentum p = ∂L
∂qdefines, for a given q, an invertible bijection between p and q, we can
define the associated Hamiltonian by
H(q,p, t) = p · q− L(q, q, t). (2.11)
The transform going from L to H is called a Legendre transform.
Example: Consider a Lagrangian based on a general quadratic kinetic energy
L(q, q) =1
2q ·M(q)q− U(q),
where M(q) is a symmetric positive definite matrix. Then p = ∂L∂q
= M(q)q. So that q =
M(q)−1p, establishing the needed bijection between p and q. In the case of the electrostaticone particle Lagrangian (2.3), we just have p = mx, which is also clearly an invertibletransform.
Theorem 1 The Euler-Lagrange equations derived from the Lagrangian L are equivalent tothe following Hamilton’s equations
dq
dt= ∇pH (2.12)
dp
dt= −∇qH (2.13)
with H(q,p, t) = p · q− L(q, q, t)
Proof. Considering that q is a function of (q,p) and using the chain rule and the definitionp = ∂L
∂q, the partial derivatives of H defined by (2.11) verify:
∂H
∂qi=
d∑j=1
(pj∂qj∂qi
)− ∂L
∂qi−
d∑j=1
∂qj∂qi
∂L
∂qj= −∂L
∂qi
∂H
∂pi= qi +
d∑j=1
(pj∂qj∂pi− ∂qj∂pi
∂L
∂qj
)= qi
where we used pj = ∂L∂qj
in both. The second equality directly yields (2.12), and the first
equation induces that the Euler-Lagrange equation dpidt
= ∂L∂qi
is equivalent to dpidt
= −∂H∂qi
,
which yields (2.13).
Let us observe that Hamilton’s equation can also be written denoting by z = (q1, . . . , qn, p1, . . . , pn),the 2n variables together
dz
dt= J∇zH, with J = Jc =
(0 In−In 0
)(2.14)
10
where In is the n-dimensional identity matrix. This is called a canonical Hamiltonian system.Then for any functional G : R2n → R, we have
dG(z(t))
dt= ∇zG ·
dz
dt= (∇zG)>J∇zH = ∇qG · ∇pH −∇pG · ∇qH.
Definition 3 The canonical Poisson bracket of two functionals F,G : R2n → R is definedby
F,G = ∇qF · ∇pG−∇pF · ∇qG.
It is then straightforward to prove the following
Proposition 1 Hamilton’s equation (2.12)–(2.13) are equivalent to
dG(z(t))
dt= G,H, ∀G ∈ C1(R2n).
Theorem 2 If the Hamiltonian does not depend explicitly on t, i.e. if ∂H∂t
= 0, it is conservedby Hamilton’s equations:
d
dtH(q,p) = 0.
Proof. Using the chain rule and (2.12)–(2.13), we find
d
dtH(q,p) =
dq
dt· ∇qH +
dp
dt· ∇pH = ∇pH · ∇qH −∇qH · ∇pH = 0.
Definition 4 The flow of a differential equation is the mapping ϕt : Rm → Rm, z(0) 7→ z(t),that maps an initial value to the corresponding solution of the differential equation.
An important aspect of Hamiltonian systems is that their flow preserve the volume.Denoting by z = (q,p), the 2n variables all together. We observe that ∇z ·(∇pH,−∇qH)> =∇q ·∇pH−∇p ·∇qH = 0. The Jacobian of the flow of the differential systems satisfying thisproperty is conserved. This is know as the preservation of volume and Liouville’s theorem.
In order to prove this theorem, we will need a Lemma on trace free matrix differentialequations
Lemma 1 Assume Z and A(Z) are m×m matrices such that trA = 0 and that
dZ
dt= A(Z)Z. (2.15)
Then d(detZ)dt
= 0. The determinant of the matrix Z is an invariant of the flow.
Proof.d
dt(detZ)(t) = lim
ε→0
detZ(t+ ε)− detZ(t)
ε.
But as Z satisfies (2.15),
Z(t+ ε) = Z(t) + εA(Z(t))Z(t) +O(ε2)
= (I + εA(Z(t)))Z(t) +O(ε2),
11
so that
detZ(t+ ε) = det(I + εA(Z(t))) detZ(t) +O(ε2)
= (1 + εtrA(Z) +O(ε2)) detZ(t) +O(ε2),
from which it follows thatd(detZ)
dt= trA(Z) = 0.
Theorem 3 (Liouville) The flow of the differential equation
dz
dt= F(z) (2.16)
conserves volume, i.e. the Jacobian detDyz, if and only if ∇ · F = 0. This implies that if ρis a solution of
∂ρ
∂t+ F · ∇zρ = 0,
then ∫ϕt(Ω)
ρ(t, z) dz =
∫Ω
ρ(t,y) dy.
Proof. The flow of the differential equation maps an initial condition y ∈ Rm to the valueat time t, that we shall denote z(t; y) to highlight the dependence on the initial conditionas well as the dependence on time. Then taking the derivative with respect to yj of the ith
component of (2.16), we find
∂2zi∂t∂yj
=∂
∂yj(Fi(z(t; y))) =
n∑k=1
∂zk∂yj
∂Fi∂zk
.
Hence the Jacobian matrix of z with respect to y denoted by Dyz verifies the differentialequation
dDyz
dt= DzF(t; y)Dyz.
Moreover trDzF(t; y) = ∇ · F = 0. This is exactly the setting of Lemma 1, so that theJacobian of the flow detDyz is conserved.
Making the change of variables z = z(t; y) = ϕt(y), whose Jacobian is one, due to theprevious relation along with detDyy = det Im = 1, we get the equality of the integrals.
A last stronger property of the Hamiltonian flow is that it is symplectic:
Definition 5 A linear transformation A : R2n → R2n is called symplectic if A>JA = J .
Theorem 4 A canonical hamiltonian flow is symplectic, i.e if z(t; y) is a solution of (2.14)with initial condition y then the linear transformation Dyz is symplectic.
12
Proof. As we saw in the previous proof Dyz is a solution of the matrix differential equation:
dDyz
dt= DzF(t; y)Dyz.
For the canonical Hamiltonian system (2.14) the right hand side is F = (∇pH,−∇qH)>, sothat
DzF(t; y) =
(∇q∇pH ∇p∇pH−∇q∇qH −∇q∇pH
).
Henced
dt(Dyz>JDyz) =
d
dt(Dyz>)JDyz +Dyz>J d
dt(Dyz).
This yields by explicit computation, using the differential equation above verified by Dyzand the explicit form of DzF(t; y) we get
JDzF(t; y) =
(−∇q∇qH −∇q∇pH−∇q∇pH −∇p∇pH
)= −∇z∇zH
so that
d
dt(Dyz)>JDyz = (Dyz)>(∇z∇zH)Dyz− (Dyz)>(∇z∇zH)Dyz = 0,
From this it follows that
(Dyz(t))>JDyz(t) = (Dyz(0))>JDyz(0) = J ,
as z(0) = y so that Dyz(0) is the identity matrix.
2.2 Motion of a particle in an electromagnetic field
2.2.1 Newton’s equations of motion
Consider a given electromagnetic field E(t,x),B(t,x). The equations of motion of a particleof charge q and mass m can be derived from Newton’s law, the acceleration being given bythe Lorentz force
dx
dt= v, (2.17)
dv
dt=
q
m(E + v ×B). (2.18)
2.2.2 Action principle
We introduce the scalar potential φ and the vector potential A such that
E = −∂A
∂t−∇φ, B = ∇× A.
The equations of motion (2.17)–(2.18) can be derived from an action principle involving thefollowing Lagrangian:
L(x, x) = (qA +1
2mx) · x− qφ. (2.19)
13
We have in this case
∇xL = q(∇xA · x−∇xφ), ∇xL = qA +mx.
So that the Euler-Lagrange equations ddt∇xL = ∇xL become at (x(t), dx
dt(t))
q(∂A
∂t+
dx
dt· ∇xA) +m
d2x
dt2= q(∇xA ·
dx
dt−∇xφ). (2.20)
Now, a straightforward computation component by component shows that
dx
dt· ∇xA−∇xA ·
dx
dt= (∇× A)× dx
dt,
so that defining v = dxdt
, (2.20) becomes
dv
dt=
q
m
(−∂A
∂t−∇xφ+ v × (∇× A)
)=
q
m(E + v ×B),
where we recognise the Newton equation (2.18).
2.2.3 Hamilton’s equations
We start by defining the generalised momentum
p = ∇xL(x, x) = qA(t,x) +mx. (2.21)
Hence x = p/m − (q/m)A(t,x) So that the Hamiltonian, in canonical coordinates (x,p) isdefined by
H(x,p) = p · x− L(x, x) = p ·( p
m− q
mA)−(qA +
1
2(p− qA)
)· 1
m(p− qA) + qφ
=1
2m(p− qA) · (p− qA) + qφ,
and Hamilton’s equations in canonical coordinates become
dx
dt= ∇pH =
1
m(p− qA), (2.22)
dp
dt= −∇xH =
q
m(∇xA) · (p− qA)− q∇xφ. (2.23)
Remark 2 We observe that when A = 0, H = |p|2/(2m) + qφ(t,x) is the sum of a termdepending only on p and a term depending only on x (not on p). Such a Hamiltonian iscalled separable, which is a very convenient property when designing numerical schemes, aswe shall see later. This is not the case in general, when A does not vanish.
14
2.2.4 The phase-space Lagrangian
Canonical coordinates have the advantage of yielding a constant symplectic structure andthe simple form of Hamilton’s canonical equations of motion, but on the other hand we loosethe physical velocity as a variable and other nice properties. In some cases it might be moreconvenient to loose the canonical form and stay with the physical velocity. This can be doneby considering, both x and v as independent variables in the Lagrangian (2.19) can then bereplaced by the so-called phase-space Lagrangian. We assume here that both potentials Aand φ are time independent
L(x,v, x, v, t) = (qA(x) +mv) · x− (1
2m|v|2 + qφ(x)). (2.24)
We notice that this corresponds to p · x − H expressed in the (x,v) variables. Moreoverthis Lagrangian does neither explicitly depend on v nor on t. In the (x,v) variables, theHamiltonian H(x,v) = 1
2m|v|2 + qφ(x) is separable.
Let us now express the Euler-Lagrange equations for q = (x,v). First as the Lagrangiandoes not depend explicitly on v, d
dt∇vL = ∇vL writes
0 = ∇vL = mx−mv,
and then ddt∇xL = ∇xL writes
d
dt(qA +mv) = q∇xA · x− q∇xφ.
This is identical to (2.20) replacing dxdt
by v, so that from the Euler-Lagrange equations, wedirectly obtain the equations of motion (2.17)–(2.18). Then, observing that ∇xH = q∇xφ,∇vH = mv and v ×B = B(x)v, with
B(x) =
0 B3 −B2
−B3 0 B1
B2 −B1 0
,
we see that
dx
dt=
1
m∇vH, (2.25)
dv
dt=
q
m2B(x)∇vH −
1
m∇xH, (2.26)
which can also be written in matrix form(dxdtdvdt
)=
(0 1
mIn
− 1mIn
qm2B(x)
)(∇xH∇vH
), or, grouping the variables
dz
dt= J (z)∇zH. (2.27)
This corresponds to a non canonical Hamiltonian system, with a Poisson bracket defined by
F,G = ∇zF · J∇zG = (∇zF )>J (∇zG). (2.28)
The matrix J (z) is called structure matrix and verifies
Ji,j = zi, zj, i, j = 1, . . . ,m. (2.29)
Definition 6 A Poisson bracket ., . is a bilinear form satisfying for any smooth functionalsF,G,H
(i) F,G = −G,F, (skew-symmetry)
15
(ii) FG,H = FG,H+GF,H, (Leibniz rule)
(iii) F,G, H+ G,H, F+ H,F, G = 0. Jacobi identity
Proposition 2 Let F,G = ∇zF · J∇zG. This defines a Poisson bracket with structurematrix J if and only if
(i) J is antisymmetric, i.e. J = −J >,
(ii) The coordinate functions verify the Jacobi identity:
zi, zj, zk+ zj, zk, zi+ zk, zi, zj = 0. (2.30)
This proposition follows directly from Proposition 6.8. of Olver’s book [17]. Section 6.1. ofthis book provides a good introduction to general Poisson brackets.
Proposition 3 The bracket defined by (2.28) is a Poisson bracket if and only if ∇ · B = 0.
Proof. Let us apply Proposition 2. The structure matrix J is clearly antisymmetric. So itremains to verify that the coordinate functions (zi), 1 ≤ i ≤ 2n satisfy the Jacobi identity.In our case, it is convenient to split the coordinates writing zi = xi for 1 ≤ i ≤ n and zi = vifor n+ 1 ≤ i ≤ 2n.
From the definition of J in (2.27), we have for any functions F,G
F,G = ∇xF · ∇vG−∇vF · ∇xG+∇vF · B∇vG. (2.31)
It follows that
xi, xj = 0, xi, vj = δi,j, vi, vj = Bi,j(x) = εi,j,kBk(x), 1 ≤ i, j ≤ 3, (2.32)
where δi,j is the Kronecker function which is 1 of i = j and 0 else, and εi,j,k is the Levi-Civitasymbol which takes the value 1 if (i, j, k) is an even permutation of (1, 2, 3), -1 if (i, j, k) isan odd permutation of (1, 2, 3) and 0 if an index appears twice.
As the two first brackets in (2.32) involving xi are constant, they will vanish, whenplugging them into another bracket, which involves derivatives. Hence, the only coordinatebrackets that remain are those involving only components of v, which we shall now computefor 1 ≤ i, j, k ≤ 3
vi, vj, vk = εi,j,kBk(x), vk = εi,j,k∂Bk
∂xk
as vi, vj = Bi,j depends only on x and the two last terms in the bracket (2.31) involvederivatives in v of this term. The non vanishing terms of the Jacobi identity (2.30) are then
vi, vj, vk+ vj, vk, vi+ vk, vi, vj = εi,j,k∂Bk
∂xk+ εj,k,i
∂Bi
∂xi+ εk,i,j
∂Bj
∂xj= εi,j,k∇ · B,
which vanishes if and only if ∇ · B = 0. We observe that the three Levi-Civita symbols arethe same as they are circular permutations of each other.
Remark 3 The Jacobi identity is the key element of the Poisson bracket. It follows imme-diately from it that if I1 and I2 are invariants of the motion, then I1, H = I2, H = 0 andthen using Jacobi’s identity with F = I1, G = I2 it follows that I1, I2, H = 0, so thatI1, I2 is also an invariant of motion.
16
2.3 The Vlasov-Maxwell equations
2.3.1 The model
The non relativistic Vlasov equation for a particle species s of charge qs and mass ms reads
∂fs∂t
+ v · ∇xfs +qsms
(E + v ×B) · ∇vf = 0. (2.33)
It is coupled non linearly to the Maxwell equations
− 1
c2
∂E
∂t+∇× B = µ0J, (2.34)
∂B
∂t+∇× E = 0, (2.35)
∇ · E = ρ/ε0, (2.36)
∇ · B = 0, (2.37)
in addition to initial and boundary conditions. Here ε0 and µ0 are the vacuum permittivityand permeability that are related to the speed of light c by µ0ε0c
2 = 1. The source for theMaxwell equation, the charge density ρ and the current density J, are obtained from theVlasov equations by
ρ =∑s
qs
∫fs dv, J =
∑s
qs
∫fsv dv. (2.38)
We shall here define an action principle for the Vlasov-Maxwell equations, based on thecharacteristics of the Vlasov equation. Let us consider the system of differential equations
dXs
dt= Vs, (2.39)
dVs
dt=
qsms
(E(t,Xs(t)) + Vs ×B(t,Xs(t))), (2.40)
with initial conditions X(0) = x0, V(0) = v0. In order to explicit the dependence of the solu-tion on the initial condition, we shall denote the solution at time t by X(t; x0,v0),V(t; x0,v0).They are called the characteristics of the Vlasov equation (2.33), and if fs,0 is the initial valueof the distribution function, the solution of the Vlasov equation can then be expressed withthe help of the characteristics
f(t,X(t; x0,v0),V(t; x0,v0)) = fs,0(x0,v0),
which means that fs is conserved along the characteristics.
2.3.2 Action principle
The phase-space Lagrangian for the motion of a single particle of mass ms and charge qs inan electromagnetic field is given in standard non canonical coordinates by
Ls(x,v, x, t) = (msv + qsA) · x− (1
2ms|v|2 + qsφ).
17
The self consistent field theory is then obtained by plugging this Lagrangian into thefollowing action proposed by Low [12]
A[Xs,Vs, φ,A] =∑
s
∫fs,0(x0,v0)Ls(Xs(t; x0,v0),Vs(t; x0,v0),
dXs
dt(t; x0,v0)) dx0 dv0 dt
+ε02
∫|∇φ+
∂A
∂t|2 dx dt− 1
2µ0
∫|∇ ×A|2 dx dt. (2.41)
where the particle distribution functions fs are taken at the initial time and the characteristicsexpress the evolution of the distribution function, and there are as many Xs, and Vs as thereare particle species.
In this variational principle the characteristics of the Vlasov equation, which are theparticles equations of motion are obtained by taking the variation with respect to X andV, the Poisson equation, is obtained by taking the variations with respect to φ and theAmpere equation is obtained by taking the variations with respect to A. The definition ofthe electromagnetic fields from the potentials
E = −∇φ− ∂A
∂t, B = ∇× A (2.42)
automatically implies the Faraday equation ∂B∂t
+∇× E = 0 and ∇ · B = 0.In order to perform the variations, we need to generalise the directional derivative we
used in the finite dimensional case to the case where the varied functions live in an infinitedimensional Banach space. This is based on the Gateaux derivative which gives a rigorousdefinition of the functional derivative used in physics for functions that are in a Banach(including Hilbert) space. Consider a functional J from a Hilbert space V into R. ItsGateaux derivative J ′, assuming it exists, is a linear form on V , which means that it mapsany function from V to a scalar. It can be computed using the Gateaux formula:
J ′[u](v) = limε→0
J [u+ εv]− J [u]
ε=
d
dε
∣∣∣∣ε=0
J [u+ εv]. (2.43)
Remark 4 In mathematics, differentiability in Banach spaces is defined based on the Frechetderivative. Let V,W be two normed spaces ( e.g. Banach spaces). A function J : U ⊂ V → Wis called Frechet differentiable at u ∈ U if there exists a bounded linear operator A : V → Wsuch that
limh→0
‖J(u+ h)− J(u)− Ah‖W‖h‖V
= 0.
Then A is the Frechet derivative of J at u and denoted by A = J ′[u]. If a function isdifferentiable, its Frechet derivative exists and then the Gateaux derivative also exists and isequal to the Frechet derivative, but there are cases where the Gateaux derivative exists butthe function is not differentiable in the sense of Frechet.
Example: Let Ω be an open subset of Rn, u ∈ H10 (Ω) and consider the functional
J [u] =1
2
∫Ω
|∇u(x)|2 dx.
18
Then for any v ∈ H10 (Ω)
J [u+ εv]− J [u]
ε=
1
2ε
∫Ω
(|∇(u+ εv)|2 − |∇u|2) dx =1
2ε
∫Ω
(2ε∇u · ∇v − ε2|∇v|2) dx
−−→ε→0
∫Ω
∇u · ∇v dx,
so that J ′[u](v) =∫
Ω∇u · ∇v dx. Here we see again that J ′[u] is an element of the dual of V
and needs to be applied to an element of v. However using Riesz’s theorem and the scalarproduct in L2(Ω), J ′[u] can be identified to an element of L2(Ω) by writing for all v ∈ Vvanishing on the boundary
J ′[u](v) =
∫Ω
∇u · ∇v dx = −∫
Ω
∆u v dx =
∫Ω
δJ
δu[u]v dx
so that δJδu
[u] = −∆u almost everywhere. δJδu
[u] is called the functional derivative of J atu. Note that the boundary terms matter. Here we have used that u ∈ H1
0 (Ω), setting theboundary terms to 0 in the integration by parts.
Remark 5 In the physics literature the Gateaux derivative is in general called the functionaldifferential or the first variation and the test function is denoted with a δ. We would thenwrite
J ′[u](v) =d
dε
∣∣∣∣ε=0
J [u+ εv] = δJ(u; δu) =
∫Ω
δJ
δu[u] δu dx, v = δu.
See for example the lecture notes by Morrison [15], Section III.
We can now compute the variations of action (2.41). Let us start with the variations withrespect to Xs, which appears only in one integral. Hence
d
dε
∣∣∣∣ε=0
A[Xs + εY,Vs, φ,A] =
∫fs,0(x0,v0)
(Y · ∇xLs +
dY
dt· ∇xLs
)dx0 dv0 dt.
Integrating in parts in time, assuming the variations vanish at initial and final time we findthe usual Euler-Lagrange equations:
d
dt∇xLs = ∇xLs. (2.44)
Applying it to the actual Lagrangian, we can perform the same computation as in (2.20)and in the same way the variations with respect to Vs yields dXs
dt= Vs, so that the equa-
tions for the characteristics obtained based on the variations of (2.41) with respect to thecharacteristics yield the classical equations of motion of a particle in an electromagnetic field
dXs
dt= Vs, (2.45)
dVs
dt=
qsms
(E + Vs ×B). (2.46)
19
Let us now come to the variations with respect to φ, plugging in directly the expressionfor the single particle Lagrangian, using also that E = −∂A
∂t−∇φ
d
dε
∣∣∣∣ε=0
A[Xs,Vs, φ+εψ,A] = −∑s
∫fs,0(x0,v0)qsψ(t,Xs(t)) dx0 dv0 dt−ε0
∫E·∇ψ dx dt.
Now using that fs is conserved along the characteristics fs,0(x0,v0) = fs(t,Xs(t),Vs(t))and performing the change of variables x = Xs(t; x0,v0), v = Vs(t; x0,v0) using that theJacobian of the transformation is one thanks to the Liouville theorem we find:∑s
∫fs,0(x0,v0)qsψ(t,Xs(t)) dx0 dv0 dt =
∑s
qs
∫fs(t,x,v)ψ(t,x) dx dv dt =
∫ρψ dx dt,
so that the variations with respect to φ vanish when
−ε0∫
E · ∇ψ dx dt =
∫ρψ dx dv dt, ∀ψ ∈ H1(Ω).
which is a weak form of the Poisson equation. An integration by parts, assuming that ψvanishes on the boundary yields the classical form ε0∇ · E = ρ.
Let us finally come to the variations with respect to A, plugging in directly the expressionfor the single particle Lagrangian, using also that E = −∂A
∂t−∇φ
d
dε
∣∣∣∣ε=0
A[Xs,Vs, φ,A + εA] =∑s
∫fs,0(x0,v0)qsA ·V(t) dx0 dv0 dt− ε0
∫∂A
∂t·E dx dt
− 1
µ0
∫∇× A · ∇ × A dx dt
Performing again the change of variables x = Xs(t; x0,v0), v = Vs(t; x0,v0), the integralover fs yields the current density and we find the weak form of Ampere’s equation:
ε0
∫∂A
∂t· E dx dt+
1
µ0
∫∇× A · ∇ × A dx dt =
∫J · A dx dt, ∀A ∈ H(curl,Ω).
Multiplying by µ0 and integrating by parts the two integrals on the left hand side assumingthat A vanishes at initial and final times as well as its tangential components on the boundarywe find the classical form of Ampere’s equation:
− 1
c2
∂E
∂t+∇× B = µ0J,
using that ε0µ0c2 = 1 and that B = ∇× A.
20
Chapter 3
Numerical methods for hamiltoniansystems
We have seen that hamiltonian systems of the form dzdt
= J (z)∇zH feature at least twoimportant conservation properties, the conservation of the Poisson bracket by the flow (see[9] (VII.4)) and the conservation of the hamiltonian H.
The flow of our Hamiltonian system is defined by the map ϕt : y 7→ Z(t; y).
Definition 7 The Poisson bracket is conserved by the flow
F ϕt, G ϕt (y) = F,G (ϕt(y)),
or equivalently(∇yZ)>J (y)(∇yZ) = J (Z(t,y)).
This corresponds to the symplecticity of the flow in the canonical case.
Definition 8 A functional C is called a Casimir of a Poisson bracket if C,F = 0 for allsmooth functions F . This is equivalent to J∇C = 0.
Proposition 4 If a flow conserves the Poisson bracket, then the Casimirs are preserved.
Proposition 5 The only flow that preserves the Poisson structure and the hamiltonian isthe exact flow.
This means that when constructing a numerical approximation, one cannot have at the sametime the exact preservation of the Poisson structure and of the hamiltonian, but approxima-tions that conserve either of them can be constructed. However when the Poisson structureis conserved it is possible to show that an approximate hamiltonian (which approximates theexact hamiltonian up to the order of the method) is exactly conserved.
3.1 Splitting methods
This section is mostly based on the review article of McLachland and Quispel [13].
21
We consider a system of differential equations where the right-hand-side can be split in asum
dZ
dt= A(t,Z(t)) = A1 + A2 + · · ·+ An (3.1)
A splitting method then consists in obtaining an approximate solution of the full systemby combining solutions of the reduced system dZi
dt= Ai for i = 1, . . . n. In the context
of geometric methods, splitting is a particularly useful tool, if each subsystem has itself thedesired geometric structure and if for each split part either the exact solution can be obtained,which is often the case in practice, or if a simpler geometric integrator can be found.
Proposition 6 (Lie-Trotter splitting) Assume that Ai is at least C1 and that the flow
can be computed exactly for each split part and denote it by ϕ(i)t (y) = Zi(t; y). Then
ψ∆t(y) = ϕ(1)∆t ϕ
(2)∆t · · · ϕ
(n)∆t (y) (3.2)
is a first order approximation of the exact solution of (3.1).
Proof. By a Taylor expansion in t, as dZi
dt= Ai, we have
ϕ(i)∆t(y) = Z(∆t; y) = y + ∆t
dZ(0; y)
dt+O(∆t2) = y + ∆tAi(0,y) +O(∆t2).
Let us now consider the case of two parts
ϕ(1)∆t ϕ
(2)∆t(y) = ϕ
(1)∆t(ϕ
(2)∆t(y)) = ϕ
(1)∆t(y + ∆tA2(0,y) +O(∆t2))
= y + ∆tA2(0,y) + ∆tA1(0,y) +O(∆t2))
= y + ∆tA(0,y) +O(∆t2) = ψ∆t(y) +O(∆t2)
By chaining the computation, the same procedure also applies for the case of n parts.
Definition 9 The adjoint ψ∗∆t of an discrete flow ψ∆t is defined by ψ∗∆t = ψ−1−∆t.
Definition 10 A numerical method is called ψ∆t symmetric (or self-adjoint or time-reversible)if ψ−∆t(ψ∆t(y)) = y.
Proposition 7 Given a method ψ∆t of order 1, the numerical method Φ∆t = ψ∆t/2 ψ∗∆t/2is symmetric and of order 2.
Proof. Using the definition of the adjoint we have
Φ−∆t(Φ∆t(y)) = ψ−∆t/2 ψ−1∆t/2 ψ∆t/2 ψ−1
−∆t/2(y) = y.
It is also clear by using the definition that Φ∗∆t = (ψ∆t/2 ψ∗∆t/2)∗ = ψ∆t/2 ψ∗∆t/2 = Φ∆t,which is the classical definition of self-adjointness.
The proof that it is second order can be found in the book by Hairer, Lubich and Wanner[9].
Remark 6 If ψ∆t = ϕ1∆t ϕ2
∆t is the composition of two exact flows verifying φ−1−∆t = φ∆t
thenψ∆t/2 ψ∗∆t/2 = ϕ1
∆t/2 ϕ2∆t/2 ϕ2
∆t/2 ϕ1∆t/2 = ϕ1
∆t/2 ϕ2∆t ϕ1
∆t/2
This is the well known Strang splitting which is of order 2 and symmetric. Symmetry isa very important property for long time integration as some errors cancel with symmetricmethods.
22
For this reason, the simplest method of choice for the splitting in n parts, is the symmetriccomposition
ψ∆t = ϕ(1)∆t/2 · · · ϕ
(n−1)∆t/2 ϕ
(n)∆t ϕ
(n−1)∆t/2 · · · ϕ
(1)∆t/2 (3.3)
Methods of arbitrary order can be derived by composition of lower order methods [9, 13].For example, the simplest fourth order method is obtained by combining the symmetricStrang-splitting ψ∆t of order 2 above to obtain
Φ(4,TJ)h = ψα∆t ψβ∆t ψα∆t, (3.4)
withα = 1/(2− 21/3), β = −21/3/(2− 21/3).
This is called the triple jump method. Note that the second coefficient β is negative (see [9],p.44).
McLachland and Quispel [13] claim that this method has a high error constant and suggestto use preferably the Suzuki fractals (see [9], p.44). The one with the least number of stageshas 5 stages and reads
Φ(4,S)h = ψα∆t ψα∆t ψβ∆t ψα∆t ψα∆t, (3.5)
withα = 1/(4− 41/3), β = −41/3/(4− 41/3).
McLachland and Quispel [13] propose what they consider some good general high-ordermethod p. 407 of their review paper, mentioning however that optimal high order methodscan be derived for some specific problems. Let us mention in particular the 6th order method
Φ(6)h = ψa1∆t ψa2∆t ψa3∆t ψa4∆t ψa5∆t ψa4∆t ψa3∆t ψa2∆t ψa1∆t, (3.6)
with a1 = 0.1867, a2 = 0.55549702371247839916, a3 = 0.12946694891347535806,a4 = −0.84326562338773460855, a5 = 1− 2
∑4i=1 ai.
3.2 Numerical integration of the motion of a charged
particle in a static electromagnetic field
Only the exact integrator conserves exactly the energy and the symplectic form. So one needsto chose. When solving numerically a Poisson system of the form
dZ
dt= J (Z)∇H,
one usually uses a splitting method, where each of the split parts has still a good structurebut is simpler to integrate, ideally each split part could be integrated exactly. For exampleif the hamiltonian can be split into several pieces that can be integrated exactly keeping thefull Poisson matrix,
dZ
dt= J (Z)∇H1 + J (Z)∇H2 + J (Z)∇H3.
we obtain a symplectic integrator.
23
Another option is to split the Poisson matrix, such that each piece remains antisymmetricand keeping the full hamiltonian
dZ
dt= J1(Z)∇H + J2(Z)∇H + J3(Z)∇H
and using a discrete gradient method, as we shall see later, one can exactly conserve H.And finally by decomposing the right-hand-side in divergence free parts, where one can
construct a volume preserving integrator for each part, one obtains globally a volume pre-serving integrator.
In our case the system reads(dxdtdvdt
)=
(0 1
mIn
− 1mIn
qm2B(x)
)(∇xH∇vH
), (3.7)
with the Hamiltonain H(x,v) = 12m|v|2 + qφ(x), and the Poisson matrix
J (x) =
(0 1
mIn
− 1mIn
qm2B(x)
).
3.2.1 Hamiltonian splitting
The above Hamiltonian can be split into H(x,v) = H1(x) + H2(v) with H1(x) = qφ(x)and H2(v) = 1
2m|v|2. Hence splitting the Hamiltonian yields the following two subsystems:
keeping only J∇H1
dx
dt= 0, (3.8)
dv
dt=
q
mE, (3.9)
and keeping only J∇H2
dx
dt= v, (3.10)
dv
dt=
q
mv ×B. (3.11)
As in the subsystem (3.8)-(3.9) E depends only on x which stays constant due to (3.8). Thissubsystem can be integrated analytically, with the solution
x(t; x0,v0) = x0, v(t; x0,v0) = v0 +q
mE(x0)t (3.12)
This is however not the case for the other subsystem (3.10)-(3.11). One option would beuse a numerical symplectic integrator for subsystem (3.10)-(3.11) or event for the full system,but this would be implicit. We won’t consider this option here.
24
3.2.2 Divergence free splitting
If we further split the system, we will not conserve the full Poisson structure in each of thesubsystems, so it will not be possible to derive a Poisson integrator. However observing thatsplitting the x components and the v components in the right-hand-side of (3.10)-(3.11) wehave
∇z ·(
v0
)= ∇x · v = 0,
and
∇z ·(
0qm
v ×B
)=
q
m∇v · (v ×B) = 0,
so that each part is divergence free and thus volume preserving. So, let us now consider alongwith (3.8)-(3.9), the further two subsystems:
dx
dt= v, (3.13)
dv
dt= 0, (3.14)
and
dx
dt= 0, (3.15)
dv
dt=
q
mv ×B. (3.16)
The analytical solution of (3.13)-(3.14) is
x(t; x0,v0) = x0 + v0t, v(t; x0,v0) = v0. (3.17)
and as x is constant along the flow, the solution of (3.16) is a rotation around the axis definedby B(x0), so that we can verify that, denoting by ω0 = q|B(x0)|/m, b0 = B(x0)/|B(x0)|,v0,‖ = v0 · b0, and by v0,⊥ = b0 × (v0 × b0), the analytical solution of (3.15)-(3.16) is
x(t; x0,v0) = x0, v(t; x0,v0) = v0,‖b0 + cos(ω0t)v0,⊥ + sin(ω0t)b0 × v0,⊥. (3.18)
Finally combining the flows, (3.12)–(3.17)–(3.18) using one of splittings proposed in theprevious section, we obtain a volume preserving approximation of (3.7). This method wasrecently investigated in [10].
On the other hand, as (3.16) is a 2D rotation, a volume preserving second order nu-merical integrator can be obtained using the implicit trapezoidal rule. Denoting by v1 =v(∆t; x0,x0), this reads
v1 − v0 =q∆t
2m(v1 + v0)×B(x0) =
ω0∆t
2(v1 + v0)× b0. (3.19)
Taking the dot product with b0, we get v1,‖ = v0,‖, then collecting v1 on the left-hand-sideand v0 on the right-hand-side, we get
v1 −ω0∆t
2v1 × b0 = v0 +
ω0∆t
2v0 × b0
25
Taking the cross product of this expression with b0 and using that v1,‖ = v0,‖ and that forany v, we have (v × b0)× b0 = −v + v‖b0 = −v⊥ , we find
v1 × b0 +ω0∆t
2v1 = v0 × b0 −
ω0∆t
2v0 + (ω0∆t)v0,‖b0.
Then, multiplying this last expression with ω0∆t2
and summing with the previous one, we get(1 +
ω20∆t2
4
)v1 =
(1− ω2
0∆t2
4
)v0 + ω0∆tv0 × b0 +
ω20∆t2
2v0,‖b0
=
(1 +
ω20∆t2
4− ω2
0∆t2
2
)v0 + ω0∆tv0 × b0 +
ω20∆t2
2v0,‖b0
=
(1 +
ω20∆t2
4
)v0 + ω0∆tv0 × b0 +
ω20∆t2
2(v0 × b0)× b0
so that we get, denoting by β = ω0∆t2
, the classical formula for the Boris push [3]
v(∆t; x0,x0) = v1 = v0 +2β
1 + β2(v0 + βv0 × b0)× b0. (3.20)
Putting it altogether, we obtain a symmetric volume preserving algorithm by combining(3.17), (3.12), and (3.20) on a time step ∆t. So starting from (xn,vn) we getFirst operator (3.17) on ∆t/2: x(1) = xn + vn∆t/2, v(1) = vn
Second operator (3.12) on ∆t/2: x(2) = x(1), v(2) = v(1) + E(x(1))∆t/2
Third operator (3.20) on ∆t: x(3) = x(2), v(3) = v(2) + 2β1+β2 (v(2) + βv(2) × b0)× b0
Second operator (3.12) on ∆t/2: x(4) = x(3), v(4) = v(3) + E(x(3))∆t/2First operator (3.17) on ∆t/2: x(5) = xn + v(4)∆t/2, v(5) = v(4).
Noticing that x does not evolve between the second and fourth step and x(1) is an ap-proximation of xn+1/2, the algorithm can be simplified and becomesstarting from (xn,vn)and indicating only the quantities that are modified at each step:
xn+1/2 = xn + vn∆t/2,
v− = vn + E(xn+1/2)∆t/2,
v+ = v− +2β
1 + β2(v− + βv− × b0)× b0,
vn+1 = v+ + E(xn+1/2)∆t/2,
xn+1 = xn+1/2 + vn+1∆t/2.
This is the classical Boris-Buneman algorithm that has been the workhorse for electromag-netic PIC for many years and although it is only second order it has much better long timeproperties than the fourth order RK4 method. This is due to the fact that it is symmetricand volume preserving as was first proved in [19].
This algorithm is very good when β = ω0∆t2
is small enough, this means when the timestep is small compared to the cyclotron period. Then the error committed by the secondorder approximation of (3.15)–(3.16) is of the order of the splitting error, else it is better touse the exact solution of (3.15)–(3.16), given by (3.18), instead. Then the middle step in theprevious algorithm is replaced by
v+ = v−‖ b0 + cos(ω0∆t)v−⊥ + sin(ω0∆t)b0 × v−⊥,
= (v · b0)b0 + b0 × (cos(ω0∆t)v− × b0 + sin(ω0∆t)v−). (3.21)
26
3.2.3 Discrete gradient methods
A general class of methods called discrete gradient methods conserves exactly a discreteHamiltonian for ODEs of the form
dz
dt= J (z)∇zH
where J is a antisymmetric matrix (not necessarily defining a Poisson bracket). They wereproposed by Mc Lachland, Quispel and Robidoux [14].
They are based on a discrete gradient ∇H(zn, zn+1) such that
(zn+1 − zn)>∇H(zn, zn+1) = H(zn+1)−H(zn),
Then the scheme zn+1−zn∆t
= J ∇H(zn, zn+1) conserves H, indeed
H(zn+1)−H(zn) = (zn+1 − zn)>∇H(zn, zn+1) = ∆t∇H>J >∇H = 0
provided J is a antisymmetric approximation of J (z).
Application to our problem: In order to find the above structure with simpler differentialequations, from (3.7) we can consider the following split parts, in which the two parts of Jremain antisymmetric (
dxdtdvdt
)=
(0 1
mIn
− 1mIn 0
)(∇xH∇vH
)= J1∇zH, (3.22)
and (dxdtdvdt
)=
(0 00 q
m2B(x)
)(∇xH∇vH
)= J2∇zH, (3.23)
the hamiltonian remaining in both cases H(x,v) = 12mv2 + qφ(x), denoting by v2 = |v|.
This hamiltonian will be conserved in each part of the splitting and so for the full composedmethod.
The second part can be written equivalently
dx
dt= 0,
dv
dt=
q
mv ×B(x).
This is the same as (3.15)–(3.16) for which an exact solution, which conserves the hamiltonianH is given by (3.18). On the other hand, the approximate solution given by (3.20) alsoconserves the hamiltonian and could also be used in an energy preserving method. Note thatas ∇xH does not appear in the final system due to the form of J2 which has 0 on the blocksof the first line and an average gradient for ∇vH is ∇vH = m
2(vn+1 + vn), so that (3.19) is
indeed a discrete gradient discretisation of (3.15)–(3.16).Let us now turn to the first subsystem (3.22), which is equivalent to
dx
dt= v,
dv
dt=
q
mE(x).
Straightforward discrete gradients can be obtained for quadratic hamiltonians, in particularas we have seen ∇vH(vn,vn+1) = m
2(vn+1 + vn), so that the first equation can be discretised
byxn+1 − xn
∆t=
1
2(vn+1 + vn) (3.24)
27
Now, the second equation of this system is more complex as it involves a general function φ(x)in the hamiltonian. In this case a general method yielding a discrete gradient is the Average-vector-field method proposed by Celledoni et al. [5]. Here the x part of the hamiltonian isqφ(x) and the second component of J1∇zH is q
mE(x) for which a discrete gradient can be
defined over a linear trajectory going from xn to xn+1 which reads
− 1
m∇xH(xn,xn+1) = q
∫ 1
0
E(xn + ζ(xn+1 − xn)) dζ.
Indeed in this case
(xn+1− xn) · (− 1
m)∇xH(xn,xn+1) = q
∫ 1
0
d
dζφ(xn + ζ(xn+1− xn)) dζ = qφ(xn+1)− qφ(xn),
So that the second discrete equation reads
vn+1 − vn
∆t= q
∫ 1
0
E(xn + ζ(xn+1 − xn)) dζ. (3.25)
Note that for the hamiltonian to be conserved, this integral needs to be computed exactly,but this can be done for example when E is a finite element approximation whose basisfunctions are polynomials on each cell. Else if φ is a smooth analytical function the integralcan be approximate to arbitrary accuracy by a high order quadrature rule. In all cases thesystem (3.24)–(3.25) needs to be solved iteratively using a Picard or a Newton method as itis nonlinearly implicit.
28
Chapter 4
Some useful notions from differentialgeometry
In this chapter we introduce the basic notions of differential geometry that will be neededfor the Finite Element Exterior Calculus. We follow mostly the definitions from the bookof Frankel [8] and the paper by Palha et al. [18] introducing so-called physics compatiblediscretization techniques.
4.1 Basic notions
The basic object of differential geometry is a manifold M:
Definition 11 A differentiable manifold of dimension n is defined by a collection of p map-pings
Fν : Uν ⊂ Rn →M,
q 7→ Fν(q)
which are at least homeomorphisms (continous, bijective and with continuous inverse), thatcoverM. The sets Uν are open and the coordinate systems are compatible in the intersectionof two such sets, and for two mappings with an intersection onM, F−1
ν Fµ is differentiable.
A useful example to understand the notion is the surface of a sphere, which is a two-dimensional manifold, that needs at least two mappings (also called charts in the differentialgeometry literature, where they are sometimes defined by the inverse of the mapping we areusing here) to be completely covered without singular point. See Figure 4.1 for an illustration.
An important goal in differential geometry is to define quantities independently of thelocal coordinates. Such quantities are called intrinsic.
Remark 7 A submanifold of dimension n of Rm, m ≥ n, is a special case of a manifold ofdimension n. In this case the manifold can be described using the cartesian coordinates ofRm by
(xν1(q), . . . , xνm(q)) = Fν(q).
29
Figure 4.1: Illustration of two charts on a sphere.
Moreover there exists locally a subset of the cartesian coordinates q = (xνσ(1), . . . , xνσ(n)),
where σ is a permutation of 1, . . . ,m, that can be used as local coordinates, and the othercomponents are functions of these:
(xνσ(n+1), . . . , xνσ(m)) = Gν(q).
Equivalently, a submanifold of dimension n of Rm can be defined by the cartesian coordinatesin Rm and a set of m− n constraints on these coordinates.
Example: A circle of radius R is a submanifold of dimension 1 of R2. It can be defined bythe constraint x2
1 +x22 = R2, by the parametrisations by an angle θ on different open intervals
x1 = R cos θ, x2 = R sin θ, θ1 < θ < θ2.
It can also be parametrised by x1 for the lower and upper part separately
x2 = ±√R2 − x2
1, −R < x1 < R,
and by x2 for the left and right part separately
x1 = ±√R2 − x2
2, −R < x2 < R.
We will use the manifold to define Finite Elements in a curvilinear coordinate system.In this case the manifold will just be Rn with in general a unique mapping F defining thecoordinate system, which is in this case a C1-diffeomorphism. Each of the mapping Fν definesa local coordinate system (which is global if there is only one mapping). From the mappingswe define the Jacobians
DFν(q) = ((∂xνi∂qj
))1≤i,j≤n, D(F−1ν )(x) = ((
∂qi∂xνj
))1≤i,j≤n.
Using the chain rule on F−1ν (Fν(q)) = q one finds that D(F−1
ν )(x) = (DFν(q))−1.
30
Tangent plane and vectors. The first important object that can be associated to a pointx in a manifold is a vector. A vector is intrinsically defined as the tangent at x to a curvepassing through x. In local coordinates a curve γ is parametrized by
γ :[a, b]→Ms 7→ (x1(q(s)), . . . xm(q(s)))
and its tangent at x = γ(c), c ∈ [a, b] is
v(c) =dγ
ds(c) =
dx
ds(c) =
(dx1
ds(c), . . . ,
dxmds
(c)
)>.
The collection of all possible curves passing through x on M define a vector for each curveand altogether the tangent plane TxM at x. If M = Rn as in our main application thetangent plane is also TxM = Rn, however this is not always the case as one can figure outby considering the surface of a sphere as a manifold.
A continuous set of vectors on a manifold defines a vector field. The collection of allpoints associated to their tangent plane
TM = (x,v), x ∈M,v ∈ TxMis called the tangent bundle.
The local coordinates defined by the mapping F also defines a basis of TxM. Indeed,using the mapping x(s) = F(q(s)), we find
dx
ds=
n∑j=1
dqj
ds
∂F
∂qj=
n∑j=1
vj∂F
∂qj.
The corresponding basis vectors are tj = ∂F∂qj
:= ∂j. They depend on x although we generally
don’t make it explicit in their expression. Geometrically the basis vector ∂i at a point x ∈Mis associated to the curves passing through x obtained by letting all qj, j 6= i, constant. Thesebasis vectors are denoted in the modern differential geometry literature by ∂j, that we willuse in these notes, or by ∂
∂qj. So that a vector v ∈ TxM can be expressed in the local
coordinate basis by
v =n∑i=1
vi∂i = (DF)v.
We denote here in serif font v, the geometrical object. One can think of it as the vectorexpressed in a reference frame, typically cartesian coordinates of Rn or Rm for an objectembedded in a higher dimension manifold (as the surface of a sphere). And we denote inbold font the components of a column vector in a local basis v = (v1, . . . , vn)>. The lastform above is a matrix vector multiplication directly following from the definition of ∂i asthe ith column of the Jacobian matrix DF. As the geometric object is independent of thechoice of local coordinates we have
v = (DF)v = (DF′)v′,
if F′ defines another local coordinate system (mapping) in which v′ = (v′1, . . . , v′n) are thecomponents of v. We follow here the convention of denoting the components of a tangentvector with upper indices and the basis vectors in the tangent plane with lower indices. Thiswill lead to the Einstein summation convention, where elements with upper index multiplyingelements with lower index are automatically summed: vi∂i =
∑ni=1 v
i∂i. Vectors or tangentvectors are also called contravariant vectors in the physics literature.
31
Example: Consider the mapping yielding polar coordinates
F(r, θ) =
(x1 = r cos θx2 = r sin θ
)A basis of the tangent plane at any point x = (x1, x2) is given by
∂r =
(cos θsin θ
), ∂θ =
(−r sin θr cos θ
).
Here ∂r is tangent to the line θ = C (only r varies) passing through x and ∂θ is tan-gent to the line r = C (only θ varies) passing through x. Note that this basis is notnormalized, as opposed to the classical basis used in polar coordinates er = (cos θ, sin θ)>,eθ = (− sin θ, cos θ)>. Moreover unlike in cartesian coordinates where the basis vectors ofthe tangent plane are constants, in polar coordinates, as in any general system of curvilinearcoordinates, or local coordinates, they depend on the point of which the tangent plane isconsidered.
4.2 Cotangent plane and covectors
Even though vectors (tangent vectors) and vector fields are the most intuitive objects fordefining dynamics on a manifold, an another object proved to be mathematically more in-teresting the covector p, which is defined as a linear form on the tangent space at x, i.e. alinear mapping from TxM→ R, so that a covector applied to a vector yields a real numberp(v) ∈ R, and is linear: p(λ1v1 + λ2v2) = λ1p(v1) + λ2p(v2). The collection of all covectorsat x is called the cotangent plane at x and denoted by T ∗xM. To a vector field we can thenassociate a covector at each corresponding point, this defines a covector field, which is moreclassically called a differential form or a 1-form. In the same way as the tangent bundle, onecan define the cotangent bundle by
T ∗M = (x,p), x ∈M,p ∈ T ∗xM .
The natural basis of T ∗xM is the dual basis of ∂1, . . . ,∂n denoted by dq1, . . . , dqn or d1, . . . , dn
and defined by di(∂j) = δij(= 1 if i = j and 0 else). From this definition, in the caseM = Rn,as the ∂j are the columns of DF and (DF)−1DF = In, the di are the lines of (DF)−1. Sothe components p> = (p1, . . . , pn) of a covector p, which is a geometric object independentof the basis are defined by
p =n∑i=1
pi di = p>(DF)−1.
The components of a covector define a row vector, which multiplies the inverse Jacobianmatrix on the left. We therefore denote them using a transpose: p>, and their componentsare denoted classically with a lower index. We can now express in local coordinates how acovector is applied to a vector:
p(v) =n∑i=1
pi di(
n∑j=1
vj∂j) =n∑i=1
pivi = p>(DF)−1(DF)v = p>v.
32
We notice that the definition of the covectors is such that their application to a vectorcan be expressed directly using any local coordinate system without using the mapping F.Indeed p(v) = p>v = p′>v′ independently of the local coordinate system. Because suchsums involving a vector and a covector arrive frequently, we will use the Einstein summationconvention denoting by
pivi =
n∑i=1
pivi.
A sum is implied where the same index appears as a lower index and an upper index in aproduct, hence implying a duality pairing between a covector and a vector.
Expressing a basis of tangent vectors in local coordinates as n column vectors of lengthm in a submanifold of dimension n of Rm, one can also use the definition of the covectors tocompute their coordinates as n row vectors of length m. The geometrical covector will againbe independent of the choice of the coordinate system.
4.3 p-forms
We have introduced 1-forms which associate a real number to any vector of the tangent planeTxM at some given point x ∈ M. We shall use an index x to denote the dependence on apoint on the manifold. We can now define the tensor product of two 1-forms, which is appliedto two vectors v and w, as
αx ⊗ βx(v,w) = αx(v)βx(w)
and the exterior product of two one forms as
αx ∧ βx(v,w) = αx ⊗ βx(v,w)− βx ⊗ αx(v,w) = αx(v)βx(w)− αx(w)βx(v).
From this definition, it follows that the exterior product of two 1-forms is antisymmetric
αx ∧ βx(v,w) = −βx ∧ αx(v,w).
The exterior product of two 1-forms should be a 2-form. So this leads to the definition ofa 2-form as an antisymmetric element of T ∗xM× T ∗xM and more generally the p-forms aredefined as follows:
Definition 12 A p-form αp is a p-linear form on TxM× · · · × TxM (p times), which isalternating, i.e.
αpx : TxM× · · · × TxM→ R
(v1, . . . ,vp) 7→ αpx(v1, . . . ,vp)
such thatαpx(. . . ,vi, . . . ,vj, . . . ) = −αpx(. . . ,vj, . . . ,vi, . . . ).
When two vectors are exchanged, all the others staying the same, the sign of the p-formchanges. This is a generalization of anti-symmetry to p variables. This implies in particular,that when a vector appears twice in the arguments the p-form vanishes. We denote a p-formby αp with an upper right index when we wish to explicit the degree p of the form. When itis useful to recall its dependency on x we write αpx.
33
Due to the alternating property, in a manifold of dimension n all p-forms for p > nvanish. This can be seen by decomposing all the vectors on a basis, which contains exactlyn-elements.
We will denote the space of p-forms on the manifoldM by Λp(M) for p = 0, . . . , n. The0-forms are the scalar functions on M. The n-forms in a manifold of dimension n are alsowell known from linear algebra. Indeed at a point x of the manifold a n-form is a n-covectordefined as an alternating n-linear form, which is the definition of the determinant of n vectors.
Definition 13 The exterior product of a k-form and a l-form is a mapping
∧ : Λk(M)× Λl(M)→ Λk+l(M) k + l ≤ n
satisfying the following properties
antisymmetry: αk ∧ βl = (−1)klβl ∧ αk (4.1)
associativity: (αk ∧ βl) ∧ γm = αk ∧ (βl ∧ γm) (4.2)
distributivity: (αk + βk) ∧ γm = αk ∧ γm + βk ∧ γm (4.3)
where αk ∈ Λk(M), βl ∈ Λl(M), γm ∈ Λm(M).
Note that the wedge product with a 0-form is just a multiplication with a scalar function.Given a basis ( dq1, . . . , dqn) of the space of 1-forms Λ1(M), a basis of Λp(M) is given by
all possible combinations of p distinct dqi by wedge product. The order is fixed arbitrarily,for example by imposing that the indices are in increasing order:
αp(x) =∑
i1<···<ip
αi1,...,ip(x) dqi1 ∧ · · · ∧ dqip .
The dimension of Λp(M) is thus
(np
)= n!/(p!(n− p)!)
Example: In R3, the zero forms are the scalar functions. The 1-forms, 2-forms, 3-formscan be expressed in the local basis by
α0(x) = α(x), (4.4)
α1(x) = α1(x) dq1 + α2(x) dq2 + α3(x) dq3, (4.5)
α2(x) = α23(x) dq2 ∧ dq3 + α31(x) dq3 ∧ dq1 + α12(x) dq1 ∧ dq2, (4.6)
α3(x) = α123(x) dq1 ∧ dq2 ∧ dq3. (4.7)
We note that the 1-forms and 2-forms have 3 scalar components and the 0-forms and 3-formshave 1 scalar component. In order to evaluate the basis p-form on vectors, one can use theformulas for determinants
dqi(v) = vi
dqi ∧ dqj(v,w) =
∣∣∣∣ dqi(v) dqi(w)dqj(v) dqj(w)
∣∣∣∣ = viwj − vjwi,
dq1 ∧ dq2 ∧ dq3(u,v,w) =
∣∣∣∣∣∣dq1(u) dq1(v) dq1(w)dq2(u) dq2(v) dq2(w)dq3(u) dq3(v) dq3(w)
∣∣∣∣∣∣ =
∣∣∣∣∣∣u1 v1 w1
u2 v2 w2
u3 v3 w3
∣∣∣∣∣∣ .34
Remark 8 Following the intuitive definition of the Riemann integral, paving the space intosmall segments, we observe that 1-forms, which are applied to one single vector, will benaturally integrated along curves, 2-forms, which take 2 vectors, will be naturally integratedon surfaces and 3-forms taking 3 vectors will be naturally integrated on volumes. 0-forms fortheir part cannot be integrated, one only takes their point values.
This idea comes back when discretising on a mesh. 0-form degrees of freedom are associ-ated to points of the mesh (e.g. vertices), 1-forms to edges, 2-forms to faces and 3-forms tocells or volumes.
4.4 Riemannian metric
The structures that we have built on our manifold up to now do not allow us to define thelength of a vector. For this we need a new structure call Riemannian metric defined asfollows:
Definition 14 A Riemannian metric g on a smooth manifold M is a scalar product (abilinear symmetric positive definite form) on the tangent spaces TxM that varies smoothlywith x, i.e.
gx : TxM× TxM→ R
such that gx is bilinear and
(i) gx(v,w) = gx(w,v), ∀v,w ∈ TxM,
(ii) gx(v,v) ≥ 0 ∀v ∈ TxM,
(iii) gx(v,v) = 0⇔ v = 0,
(iv) For any smooth vector fields vx,wx, the mapping x 7→ gx(vx,wx) is smooth.
In our applications, the tangent planes are Rn, which is equipped with the euclidian scalarproduct ·. We can express it in the natural basis of TxM, ∂1, . . . ,∂n:
gx(v,w) = v · w =n∑
i,j=1
viwj∂i · ∂j.
Denoting by gij = ∂i · ∂j, the scalar product is expressed in the tangent basis ∂1, . . . ,∂n by
gx(v,w) =n∑
i,j=1
viwjgij = viwjgij
using the Einstein summation convention. Obviously this reduces to the classical formula∑viwi when the canonical basis of Rn is used (corresponding to an identity mapping) as
gij = δij in this case.The metric enables us to associate a covector to a vector in a unique way. Indeed, given
a vector v ∈ TxM, w 7→ gx(v,w) defines a linear form on TxM, i.e. an element p ∈ T ∗xMsuch that p(w) = gx(v,w). This can be written in local coordinates
p>w = v>Gw or equivalently pjwj = vigijw
j
35
using Einstein’s summation convention, that we will use systematically in the sequel, whereG is the n × n matrix with entries gij. As this is verified for any w ∈ TxM, it follows thatpi = gijv
j (⇔ p> = v>G). In particular in cartesian coordinates, where G is the identitymatrix, a covector has the same components as the vector it is associated to. Note that dueto the definition of gij with the tangent vectors, G can be expressed using the Jacobian of themapping G = (DF)>DF and the inverse of G whose coefficients are denoted by gij verifiesG−1 = (DF)−1DF−>, so that
gij = di · dj. (4.8)
4.5 Exterior derivative
Definition 15 The exterior derivative, denoted by d is the unique operator from Λp(M)→Λp+1(M) satisfying:
(i) d is additive: d(α + β) = dα + dβ,
(ii) dα0 is the usual differential of functions,
(iii) d(αp ∧ βq) = dαp ∧ βq + (−1)pαp ∧ dβq, (Leibniz rule),
(iv) d2α = d( dα) = 0.
See Frankel [8] p. 53 for a proof of the uniqueness.Let us compute exterior derivatives in R3. For this we shall in particular need d dqi = 0
and (iii) for a scalar function, which is a 0-form (p = 0):
d(α0 ∧ βq) = dα0 ∧ βq + α0 dβq.
For a 0-form α0x = α(q) for some smooth scalar function α, we get
dα0x =
∂α
∂q1(q) dq1 +
∂α
∂q2(q) dq2 +
∂α
∂q3(q) dq3.
For a 1-form α1x = α1(q) dq1 + α2(q) dq2 + α3(q) dq3, we have
dα1x = dα1(q) ∧ dq1 + dα2(q) ∧ dq2 + dα3(q) ∧ dq3
=3∑i=1
(∂αi∂q1
(q) dq1 +∂αi∂q2
(q) dq2 +∂αi∂q3
(q) dq3
)∧ dqi
=
(∂α2
∂q1− ∂α1
∂q2
)dq1 ∧ dq2 +
(∂α1
∂q3− ∂α3
∂q1
)dq3 ∧ dq1
+
(∂α3
∂q2− ∂α2
∂q3
)dq2 ∧ dq3
Note that the components of the exterior derivative of the 1-form are the components of thecurl in cartesian coordinates.
For a 2-form α2(x) = α23(q) dq2 ∧ dq3 + α31(q) dq3 ∧ dq1 + α12(q) dq1 ∧ dq2, we have
dα2(x) = dα23 ∧ dq2 ∧ dq3 + dα31 ∧ dq3 ∧ dq1 + dα12 ∧ dq1 ∧ dq2
=
(∂α23
∂q1+∂α31
∂q2+∂α12
∂q3
)dq1 ∧ dq2 ∧ dq3.
Here we recognize the formula for the divergence in cartesian coordinates. And finally for a3-form α(3)(x) = α123(q) dq1 ∧ dq2 ∧ dq3, we get dα(3) = 0.
36
4.6 Interior product
Definition 16 The interior product of a p-form αp and a vector v is the (p−1)-form denotedby ivα
p such thativα
p(v1, . . . ,vp−1) = αp(v,v1, . . . ,vp−1). (4.9)
For a 0-form ivα0 = 0.
Proposition 8 The interior product iv : Λp(M) → Λp−1(M) is an anti-derivation, i.e. itverifies the Leibniz rule
iv(αp ∧ βq) = (ivαp) ∧ βq + (−1)pαp ∧ (ivβ
q). (4.10)
Note that this is the same property as property (iii) of the exterior derivative which isalso an anti-derivation. The difference between an anti-derivation and a standard derivation(property of derivatives) is the (−1)p factor in front of the second term.
4.7 Orientation and twisted forms
In an n-dimensional vector space two bases are related by a transformation matrix P suchthat, detP 6= 0. From that one can define two classes of bases, those that have the sameorientation are related by a transformation matrix with a positive sign and those that havea different orientation by a transformation matrix with a negative sign. The orientation of avector space is chosen by arbitrarily choosing a basis, which defines the positive orientation.This is classically done in R3 with the right-hand rule.
Definition 17 We say that a n-dimensional manifold Mn is orientable, if it can be coveredwith coordinate patches having all the same orientation.
A classical example of non orientable manifold is the moebius band.We shall denote by o(∂1, . . . ,∂n) the orientation of a basis defined by a local coordinate
system, o(∂1, . . . ,∂n) can take the values +1 or −1.The orientation is used to define twisted forms sometimes also called pseudo-forms, which
are defined not to depend on the orientation of the local coordinate system. To achieve this,the orientation is part of the definition of the twisted-form: In a positively oriented localcoordinate system αp = αp and in a negatively oriented local coordinate system αp = −αp.
An important twisted form is the volume form, which is a twisted n-form represented ina local coordinate system (q1, . . . , qn) on a riemannian manifold of dimension n by
voln = o(∂1, . . .∂n)√g dq1 ∧ · · · ∧ dqn,
where√g = det(DF) is the Jacobian of the mapping defining the local coordinate system.
Indeed as G = (DF)>DF, g := detG = (det(DF))2.
Remark 9 As the metric can be used to associate a 1-form to a vector, the volume formalong with the interior product defines a (n-1)-form associated to a vector: Given a vector vivvoln defines a (n-1)-form.
In particular in a three dimensional manifold for a vector u
iuvol3(v,w) = vol3(u,v,w) =√g
∣∣∣∣∣∣u1 v1 w1
u2 v2 w2
u3 v3 w3
∣∣∣∣∣∣37
4.8 Scalar product of p-forms and Hodge ? operator
Given a metric g on an n-dimensional manifold M, we can define a scalar product of theexterior product of p 1-form basis functions in the following way ([11] Section 3.3):
〈 dqi1 ∧ · · · ∧ dqip , dqj1 ∧ · · · ∧ dqjp〉 = det( dqim · dqjn)1≤m,n≤p = det(gim,jn)1≤m,n≤p, (4.11)
where the last equality follows from the definition of the metric (4.8). Now for two genericp-forms written in local coordinates
αpx =∑
i1<···<ip
αi1...ip dqi1 ∧ · · · ∧ dqip , βpx =∑
j1<···<jp
βj1...jp dqj1 ∧ · · · ∧ dqjp ,
it follows by multilinearity, that we define the scalar product of two p-forms at x on TxM by
〈αpx, βpx〉 =∑
i1<···<ip
αi1...ipβj1...jp〈 dqi1 ∧ · · · ∧ dqip , dqj1 ∧ · · · ∧ dqjp〉
=∑
i1<···<ip
αi1...ipβj1...jp det(gim,jn)m,n, (4.12)
=∑
i1<···<ip
αi1...ipβi1...ip . (4.13)
The last expression comes from the appropriate definition of index raising for p-forms asfollows
βi1...ip =∑
k1,...,kp
gi1k1 . . . gipkpβk1...kp . (4.14)
Beware that here, in order to capture all the terms in the determinant, the sum is over allpossible k1, . . . , kn, not only those in increasing order. For a permutation σ of the indicesβσ(k1...kp) = sgn(σ)βk1...kp , sgn(σ) being 1 for an even permutation and -1 for an odd per-mutation, so that all needed terms can be defined from those in increasing order that areknown. For repeated indices the coefficient is zero. We observe that for 1-forms this yieldsthe classical index raising formula.
We now define the Hodge ? of αp at x and denote by ?αpx the pseudo (n− p)-form
?αpx = αn−px =∑
j1<···<jn−p
α∗j1...jn−pdqj1 ∧ · · · ∧ dqjn−p ,
with α∗j1...jn−p=√|g|αk1...kpεk1...kpj1...jn−p using the index raising formula (4.14) and where
εi1...in assumes each index appears only once and εi1...in = 1 for an even permutation of(1, . . . , n) and εi1...in = −1 for an odd permutation.
As an example, let us compute the Hodge ? of general 1-forms and 2-forms in cartesiancoordinates:
?(α1 dq1 + α2 dq2 + α3 dq3) = α1 dq2 ∧ dq3 + α2 dq3 ∧ dq1 + α3 dq1 ∧ dq2.
An important example is the Hodge of the constant 0-form 1, which is the twisted volumeform voln:
?1 =√|g|ε12...n dq1 ∧ · · · ∧ dqn = voln.
38
Note also that the Hodge ? of a twisted p-form defines a (n-p)-form.Let us also recall that applying twice the Hogde operator to a p-form in a manifold of
dimension n yields? ? αp = (−1)p(n−p)αp. (4.15)
We observe that for n odd, in particular for n = 3, we always have ? ? αp = αp.From this definition of the Hodge ? it follows that for two p-forms αp and βp
αp ∧ ?βp =∑
k1<···<kp
αk1...kpβk1...kpvoln = 〈αp, βp〉voln,
using the scalar product of p-forms (4.13).Integrating over the manifold we can then define the following scalar product on Λp(M)
(αp, βp) =
∫〈αpx, βpx〉voln =
∫αp ∧ ?βp. (4.16)
Given this scalar product and its associated norm we define the Hilbert spaces
L2Λp(M) = αp ∈ Λp(M) | (αp, αp) < +∞,
HΛp(M) = ωp ∈ L2Λp(M), dωp ∈ L2Λp+1(Ω).These Hilbert spaces of differential forms form the so-called de Rham complex, which is
at the heart of the FEEC theory
0→ HΛ0(M)d→ HΛ1(M)
d→ · · · d→ HΛn(M)→ 0. (4.17)
In a three dimensional manifold the de Rham complex can also be express in the classicalvector analysis language:
0→ H1(Ω)grad→ H(curl,Ω)
curl→ H(div,Ω)div→ L2(Ω)→ 0.
4.9 Pullback of differential forms
Consider a smooth map ϕ :M→M′, where M and M′ are two manifolds, not necessarilyof the same dimension.
Through this map a vector in the tangent plane v ∈ TxM induces a vector ϕ∗v = (Dϕ)v ∈Tϕ(x)M′ in the tangent plane at ϕ(x) ofM′ called the push-forward of v. From this, one canfor a p-form ωx′ at a point x′ ∈M′ associate a p-form at the pre-image of x′ onM denotedby ϕ∗ω and called pull-back of the p-form ω. It is defined by
(ϕ∗ω)x(v1, . . . ,vp) = ωϕ(x)((Dϕ)xv1, . . . , (Dϕ)xvp). (4.18)
Very important and convenient properties of the pullback are that they respect exteriorproducts and differentiation
ϕ∗(α ∧ β) = (ϕ∗α) ∧ (ϕ∗β), d(ϕ∗α) = ϕ∗( dα). (4.19)
An important special case is when M′ is a submanifold of M and ϕ the inclusion, thenthe pullback is the trace map. With this definition we can write the Stokes theorem∫
Mdω =
∫∂M
trω.
39
Green formula and codifferential operator Using the Leibniz rule, (iii) of definition8, and the Stokes theorem, we can derive a Green type integration by parts formula fordifferential forms∫
Mdαp−1 ∧ βn−p = (−1)p
∫Mαp−1 ∧ dβn−p +
∫∂M
trαp−1 ∧ tr βn−p,
∀αp−1 ∈ Λp−1(M), βn−p ∈ Λn−p(M). (4.20)
Let us now introduce the coderivative operator d∗ : Λp(M)→ Λp−1(M) defined by
?d∗αp = (−1)p d?αp. (4.21)
We can then express the Green formula (4.20) using the scalar product on differential forms(4.16). Indeed setting βn−p =?γp, (4.20) becomes∫
Mdαp−1 ∧?γp = (−1)p
∫Mαp−1 ∧ d?γp +
∫∂M
trαp−1 ∧ tr ?γp,
and so using the definition of the codifferential and of the scalar product (4.16) this can bewritten equivalently
( dαp−1, γp) = (αp−1, d∗γp) +
∫∂M
trαp−1 ∧ tr ?γp. (4.22)
Hence, the codifferential is the formal adjoint of the exterior derivative. It is the adjoint ifthe boundary terms vanish. This explains the notation.
4.10 Maxwell’s equations with differential forms
There is a natural interpretation of physics objects with differential forms. In many casesit is more natural than with vectors. For example a force is naturally defined through thework it performs along a curve, which leads to its being a 1-form. The magnetic intensity ismeasured as a flux through a loop and thus is naturally a 2-form.
In classical vector analysis, Maxwell’s equations read
−∂D
∂t+∇× H = J, (4.23)
∂B
∂t+∇× E = 0, (4.24)
∇ · D = ρ, (4.25)
∇ · B = 0. (4.26)
These relations are supplemented by the material constitutive relations D = εE, B = µH,where ε is the permittivity tensor (in vacuum it is the constant ε0), and µ is the permeabilitytensor of the material (in vacuum it is the constant µ0). H and E are respectively the mag-netic field and the electric field, whereas B is the called magnetic flux intensity or magneticinduction and D is the electric displacement field.
40
Given that the curl corresponds to the exterior derivative applied to a 1-form and thedivergence to the exterior derivative applied to a 2-form, we find a natural way of writingMaxwell’s equations in terms of differential forms
−∂d2
∂t+ dh1 = j2 (4.27)
∂b2
∂t+ de1 = 0, (4.28)
dd2 = ρ3, (4.29)
db2 = 0. (4.30)
The constitutive relations associate 2-forms to 1-forms and thus need to be described by aHodge ? operator in the differential forms description:
d2 = ?e1, h1 = ?b2. (4.31)
As the Hodge operator associates twisted forms to straight forms and conversely, it followsfrom this that either the forms in Ampere’s equations (4.27) are twisted forms and the formsin Faraday’s equation (4.28) are straight forms other the other way. Mathematically bothoptions are possible, but physically the sources ρ3 and j2 are charge and current densitieswhich should be twisted see e.g. [4, 7]. Therefore, we will consider that h1, d2, j2 and ρ3
are twisted forms and e1 and b2 are straight forms. The electrostatic scalar potential φ andthe vector potential A also play an important role in electromagnetism, they can be describerespectively by the 0-form φ0 and the 1-form a1 and are then related to the electric field andmagnetic induction by
e1 = − dφ0 − ∂a1
∂t, b2 = da1. (4.32)
Note that taking the exterior derivative of the first equation yields Faraday’s law (4.28):
de1 = − ddφ0 − ∂ da1
∂t= −∂b2
∂t,
and taking the exterior derivative of the first equation yields (4.30)
db2 = dda1 = 0.
Another important physical object related to Maxwell’s equations is the Lorentz force in-volving E + v ×B. The velocity vector being a vector, this quantity is naturally expressedin the language of differential forms as e1 − ivb2.
Remark 10 It is important to note here that the exterior derivative and the time derivativehere are independent of the metric and thus Maxwell’s equations (4.27)-(4.30) as well asthe equation defining the potentials (4.32) have the same form in any coordinate system.The metric only appears in the Hodge operator (4.31), which relates the 1-form and 2-formexpressions of the electric and magnetic fields.
A geometric discretisation should aim at preserving these properties. There are at leasttwo classical strategies to discretise the geometric form of Maxwell’s equations. The first oneis based on Finite Difference in the framework of the so-called Discrete Exterior Calculus
41
(DEC) [6]. In this framework dual grids are constructed, the straight forms being discretisedon the primal grid and the twisted forms on the dual grid. The discrete Hodge operatorsdefines a projection from one grid to the other. The second one is called Finite ElementExterior Calculus (FEEC) [2, 1]. It is constructed on compatible Finite Elements spaces foreach degree of forms and a unique grid. The straight forms are approximated in a strong sense,yielding relations between the coefficients in the basis expressions of the discrete differentialforms. The twisted forms are approximated in the same Finite Element spaces but in weakform. This can be naturally expressed using the codiferential. Indeed using (4.31), Ampere’sequations can also be written
−∂?e1
∂t+ d?b2 = j2
and using that ?? = Id in three dimensions
−∂e1
∂t+ d∗b2 = ?j2.
In the same way we can express (4.29) using e1: d ? e1 = ρ3 and using the expression of thecodifferential for a 1-form
− d∗e1 = ?ρ3.
The expression of Maxwell’s equations with differential forms suitable for Finite Elementdiscretisation thus reads
−∂e1
∂t+ d∗b2 = ?j2 (4.33)
∂b2
∂t+ de1 = 0, (4.34)
− d∗e1 = ?ρ3, (4.35)
db2 = 0. (4.36)
The two equations involving the codifferential can be expressed in weak form to replacethe codifferential by the exterior derivative. Indeed taking the scalar product of (4.33) witha test 1-form f1 and using that d∗ is the adjoint of d provided the boundary terms vanish(else they need to be added as required by (4.22))
− d
dt(e1, f1) + (b2, df1) = (?j2, f1) ∀f1 ∈ HΛ1(M). (4.37)
We recognize here the weak form of Ampere’s equation and in the same way taking the innerproduct of (4.35) with a test 0-form yields the weak form of Gauss’s law:
−(e1, dψ0) = (?ρ3, ψ0) ∀ψ0 ∈ HΛ0(M). (4.38)
42
Chapter 5
Finite Element Exterior Calculus(FEEC)
5.1 The Finite Element Method
The Finite Element method is a classical method for finding an approximate solution ofa PDE. It consists in deriving a variational formulation of the PDE, by rewriting it as aminimisation problem, or more generally by multiplying it by a test function and possiblyintegrating it by parts to obtain a well posed solution in some well-chosen function space.The next step is to define a Finite Dimensional approximation of the function space, generallywith simple functions, like piecewise polynomials, and replace in the variational formulationthe functions in the infinite dimensional function space by their approximations in the finitedimensional space. Let us apply this to Poisson’s equation:
−∆φ = ρ in Ω, φ = 0 on ∂Ω. (5.1)
We then multiply the equation by a smooth function ψ and integrate over the domain Ω ⊂ Rn,using the Green formula −
∫Ω
∆φψ dx =∫
Ω∇φ · ∇ψ dx−
∫∂Ω
∂φ∂νψ dσ, which yields, taking
ψ = 0 on ∂Ω. ∫Ω
∇φ · ∇ψ dx =
∫Ω
ρψ dx ∀ψ ∈ H10 (Ω). (5.2)
Using the Lax-Milgram theorem one can prove that this variational formulation is well posedin H1
0 (Ω). Now the Finite Element (or Galerkin) Method consists in considering Vh =span(ϕ1, . . . , ϕN) ⊂ V and approximate the unknown φ as well as the test functions ψby elements of Vh.
The Poisson equation can also be expressed by using only first order differential operatorand introducing the additional unknown function u = −∇φ. Then (5.1) is equivalent to
div u = ρ, u = −∇φ in Ω, φ = 0 on ∂Ω. (5.3)
We now need a variational formulation involving a space for u and a space for φ. One ofthese two equations can be made to hold strongly, i.e. almost everywhere in L2, the other willbe verified weakly. This yields the following variational formulations:
43
1. Find (φ,u) ∈ H10 (Ω)×H(curl,Ω) such that∫
Ω
u · v dx +
∫Ω
∇φ · v dx = 0 ∀v ∈ H(curl,Ω), (5.4)
−∫
Ω
u · ∇ψ dx =
∫Ω
ρψ dx ∀ψ ∈ H10 (Ω). (5.5)
Note that taking v = ∇ψ in (5.4) and adding to (5.5) yields (5.2). This can bereproduced at the discrete level provided the gradients of the Finite Element subspaceof H1
0 (Ω) are in the Finite Element subspace of H(curl,Ω).
2. Find (φ,u) ∈ L2(Ω)×H(div,Ω) such that∫Ω
u · v dx−∫
Ω
φ div v dx = 0 ∀v ∈ H(div,Ω), (5.6)∫Ω
div uψ dx =
∫Ω
ρψ dx ∀ψ ∈ L2(Ω). (5.7)
In this case div u = ρ is satisfied strongly and u = −∇φ weakly. Note also that φ = 0on ∂Ω is verified as a natural boundary condition in (5.6). This formulation is knownas the mixed Poisson formulation.
5.2 The de Rham complex
For the mixed Finite Element method, compatible Finite Element spaces need to be chosenfor the method to be well-defined and stable. This can be obtained by taking Finite Ele-ment spaces defining a discrete de Rham complex, each of the spaces being a subset of thecorresponding continuous space. In standard vector calculus language this reads
H1(Ω) H(curl,Ω) H(div,Ω) L2(Ω)
V0 V1 V2 V3
Π0
grad
Π1
grad
Π2 Π3
curl
curl
div
div
(5.8)
The first row of (5.13) represents the exact sequence of function spaces involved in Maxwell’sequations in the sense that at each node, the kernel of the next operator is the image of theprevious: Im(grad) = Ker(curl), Im(curl) = Ker(div). The bottom row of (5.13) defines anexact sequence of finite dimensional function spaces. Moreover the diagram is commuting ateach level, e.g. for φ ∈ H1(Ω):
Π1(gradφ) = grad(Π0 φ).
The de Rham complex can also be expressed in the language of differential forms for anarbitrary dimension n as we have already seen in the continuous case
HΛ0(M) HΛ1(M) · · · HΛn(M)
V0 V1 · · · Vn
Π0
d
Π1
grad
Πn
d
d
d
d
(5.9)
44
Classical Finite Element methods are generally constructed element by element, whichare generally triangles of quads in 2D and tetrahedra or cubes in 3D. A Finite Elementapproximation is defined by a finite dimensional function space with dimension N , and Ndegrees of freedom which characterise uniquely an element of the previous finite dimensionalfunction space. The simplest examples are the Lagrange Finite Elements for scalar functionsschematically represented for triangles on the left where the Finite Dimensional space is
Pk = Span(xαyβ), (α, β) ∈ N2, 0 ≤ α + β ≤ k,
for k = 3, the degrees of freedom being point values of the function at the specified points.For quads the Finite Dimensional space is
Qk = Span(xα11 . . . xαn
n ), (α1, . . . , αn) ∈ Nn, 0 ≤ α1, . . . , αn ≤ k
here for k = 2.
We also end up with a total of 6 basis functions, with each basis having a value 1 at its assigned node and avalue of zero at all the other mesh nodes in the element.
Again this has the form
so we can easily determine the coefficients of all 6 basis functions in the quadratic case by inverting the matrixM. Each column of the inverted matrix is a set of coefficients for a basis function.
P3 Cubic ElementsP3 Cubic Elements
In the case of cubic finite elements, the basis functions are cubic polynomials of degree 3 with the designationP3
We now have 10 unknowns so we need a total of 10 node points on the triangle element to determine thecoefficients of the basis functions in the same manner as the P2 case.
Experiment 1Experiment 1In this experiment we will solve Poisson’s equation over a square region using Finite Elements with linear,quadratic and cubic elements and compare the results. We will solve
Similarly, here are the interpolation functions of the nine-node Lagrange element. The nodes in the x-direction are placed at x = , x = 0, and x
= while the the nodes in the y-direction are at y = , y = 0, and y = .
In[28]:=Transpose[HermiteElement1D[-L1,0,L1,0,x]].HermiteElement1D[-L2,0,L2,0,y]
Out[28]=
However, the situation becomes more complicated when you consider the elements whose nodes are not equally spaced in a grid. For example,
you cannot generate the shape functions for triangular elements or the rectangular elements with irregular node locations by using the above
tensor product approach. Here are two examples in which the interpolation functions cannot be generated.
In[29]:=p1=ElementPlot[1,-1,-1,-1,0,3/2,0,0, AspectRatio->1, Axes->True, PlotRange->-2,2,-2,2, NodeColor->RGBColor[0,0,1], DisplayFunction->Identity];
In[30]:=p2=ElementPlot[1,1,1,-1,-1,-1,-1,1,0,0, AspectRatio->1, Axes->True, PlotRange->-2,2,-2,2, NodeColor->RGBColor[0,0,1], DisplayFunction->Identity];
In[31]:=
5.3.2 Triangular Elements
Use the functions TriangularCompleteness, RectangularCompleteness, and ShapeFunction2D to generate interpolation
functions for triangular and general rectangular elements.
First, consider the triangular element with four nodal points and obtain a set of Lagrange-type shape functions, namely c = 0. Do not specify the
minimum number of symmetric terms to be dropped from the full polynomial.
Figure 5.1: (Left) P3 Finite Element, (Right) Q2 Finite Element
From the definition of the degrees of freedom, the single elements are glued togethersharing the degrees of freedom on the interface, such that the global Finite Element space,e.g. Pk(Ω) is continuous and hence a subspace of H1(Ω).
In order to define subspaces of H(curl,Ω) and H(div,Ω), for which only respectively thetangential and the normal components need to be continuous at the interface between twoelements the degrees of freedom need to be defined accordingly. Typical spaces, which forma discrete de Rham complex, with the Lagrange Finite Element as the first space are in 2Dthe Nedelec space Nk ⊂ H(curl,Ω), the Raviart-Thomas space RTk ⊂ H(div,Ω) and thediscontinous Galerkin space DGk−1 ⊂ L2(Ω), which is completely discontinuous.
The commuting diagram defined in (5.13) is the essential piece needed in the FEEC theoryto ensure stability and convergence of the Finite Element approximation. These complexesare described in [2, 1]. A classical complex based on polynomials of degree k for a mesh oftetrahedra is
0→ Pk(Ω)grad→ Nk(Ω)
curl→ RTk(Ω)div→ DGk−1(Ω)→ 0.
5.3 Discrete differential forms based on B-splines
5.3.1 B-Splines
We will consider here mostly the Isogeometric Analysis point of view, where we have aphysical mesh mapped from a cartesian logical mesh (see Figure 5.2). An important insight
45
for Finite Elements is to understand what properties depend only on the connectivity ofthe mesh (topological properties invariant by continuous deformation of the mesh) and whatproperties depend on the mapping (geometrical properties).
Figure 5.2: An example of a mapping from the logical to the physical mesh
Let us now construct a Finite Element discretisation using basis functions for the differentspaces based on B-Splines.
In order to define a family of n B-splines of degree k, we need (xi)06i6n+k a non-decreasingsequence of points on the real line called knots in the spline terminology. There can be severalknots at the same position. In the case when there are m knots at the same point, we saythat the knot has multiplicity m.
Definition 18 (B-Spline) Let (xi)06i6n+k be a non-decreasing sequence of knots. We thendefine n functions called B-Splines, where the j-th B-Spline (0 ≤ j ≤ n − 1) denoted by Nk
j
of degree k is defined by the recurrence relation:
Nkj (x) = wkj (x)Nk−1
j (x) + (1− wkj+1(x))Nk−1j+1 (x)
where,
wkj (x) =x− xj
xj+k − xj, N0
j (x) = χ[xj ,xj+1[(x).
We note some important properties of a B-splines basis:
• B-splines are piecewise polynomial of degree k,
• B-splines are non negative,
• Compact support; the support of Nkj is contained in [xj, .., xj+k+1],
• Partition of unity:∑n−1
i=0 Nki (x) = 1,∀x ∈ R,
• Local linear independence,
• If a knot xi has a multiplicity m then the B-spline is C(k−m) at xi.
A key point for constructing a complex of Finite Element spaces for p-forms comes fromthe recursion formula for the derivatives:
Nki
′(x) = k
(Nk−1i (x)
xi+k − xi−
Nk−1i+1 (x)
xi+k+1 − xi+1
). (5.10)
46
Figure 5.3: All B-splines functions associated to a knot sequence defined by n = 9, k = 2,T = 000, 1
414, 1
212, 3
434, 111
It will be convenient to introduce the notation Dki (x) = k
Nk−1i (x)
xi+k−xi. Then the recursion
formula for derivatives simply becomes
Nki
′(x) = Dk
i (x)−Dki+1(x). (5.11)
Remark 11 In the case where all knots, except the boundary knots are of multiplicity 1,the set (Nk
i )0≤i≤n−1 of B-splines of degree k forms a basis of the spline space (dimSk = n)defined by
Sk = v ∈ Ck−1([x0, xn]) |v|[xi,xi+1] ∈ Pk([xi, xi+1]).
The boundary knots are chosen to have multiplicity k + 1 so that the spline becomes interpo-latory on the boundary in order to simplify the application of Dirichlet boundary conditions.This setting is called open boundary conditions for the spline.
Then due to the definitions if follows immediately that (Dki )1≤i≤n−1 is a basis of Sk−1.
Note that if the first knot has multiplicity k+1, Dk0 will have a support restricted to one point
and be identically 0.
Remark 12 Splines can be easily defined in the case of periodic boundary conditions bytaking a periodic knot sequence. The dimension of the spline space is then the number ofdistinct knots.
5.3.2 Discrete differential forms in 1D
In 1D the continuous and its corresponding discrete de Rham complexes are defined by
H1(Ω) L2(Ω)
V0 V1
Π0
grad
Π1
grad
(5.12)
where in 1D, we denote by gradψ = dψdx
= ψ′(x). For the first line of the diagram, we dealwith classical continuous functions, so for ψ ∈ H1 we obviously have gradψ ∈ L2.
47
Let us now construct V0 and V1, such that for ψh ∈ V0 we have gradψh ∈ V1. We startwith the space of discrete 0-forms as being the spline space V0 = Sk (of dimension n) andwrite any function ψh ∈ V0 as
ψh(x) =n−1∑j=0
c0jN
kj (x).
Now for ψh ∈ V0 we take its gradient (or exterior derivative), then
ψ′h(x) =n−1∑j=0
c0j(N
kj )′(x) =
n−1∑j=0
c0j(D
kj (x)−Dk
j+1(x)) =n−1∑j=1
(c0j − c0
j−1)Dkj (x)
using (5.11), a change of index and the fact thatDk0 = 0. Then defining V1 = Span(Dk
j )1≤j≤n−1,we have that for any ψh ∈ V0, gradψh ∈ V1.
The commuting projectors: We first construct the Π0 projector for 0-forms. As 0-formsare characterised by point values, the natural projector is the interpolation at a set of ndistinct points for the spline space V0 of dimension n. The classical interpolation points forB-Splines are the Greville points, which are defined from the knots x0 ≤ x1 ≤ · · · ≤ xn+k
defining the splines of degree k by
ξi =xi+1 + · · ·+ xi+k
k0 ≤ i ≤ n− 1.
This becomes for k = 1, ξi = xi+1, for k = 2, ξi = (xi+1 + xi+2)/2 and for k = 3, ξi =(xi+1 + xi+2 + xi+3)/3. As xi denotes the point at the left of the support of the B-SplineNki (x), we notice that the Greville point ξi is located at the maximum of this B-Spline. This
remains true for higher order B-Splines.Consider now an arbitrary function ψ ∈ H1, then its interpolant at the Greville points
ψh(x) = (Π0ψ)(x) is defined uniquely by
σ0i (ψh) = ψh(ξi) =
n−1∑j=0
c0jN
kj (ξi) = ψ(ξi) = σ0
i (ψ) ∀ψ ∈ H1 0 ≤ i ≤ n− 1,
where we denote by σ0i (ψ) the Finite Element degrees of freedom, which are the linear forms
characterising an element of the Finite Element space V0. We shall denote by σ0(ψ) the vectorwhose components are σ0
i (ψ). The matrix Ik0 = (Nkj (ξi))0≤i,j≤n−1 is the spline interpolation
matrix of degree k at the Greville points. This is an invertible matrix which defines uniquelythe spline coefficients (c0
j) characterising ψh = Π0ψ from the values of ψ at the Grevillepoints.
Let us now construct the projection Π1 in order to get the commuting diagram grad Π0 =Π1grad . As this operator applies to 1-forms (in 1D), the natural degrees of freedom are cellintegrals, which should be, in order to get the commuting diagram with the definition ofΠ0 based on interpolation at the Greville points, the histopolation between two successiveGreville points. This means that for any ϕ ∈ L2, its histopolant ϕh(x) = (Π1ϕ)(x) is uniquelycharacterised by
σ1i (ϕh) =
∫ ξi
ξi−1
ϕh(x) dx =n−1∑j=1
c1j
∫ ξi
ξi−1
Dkj (x) dx = σ1
i (ϕ) for 1 ≤ i ≤ n− 1.
48
The matrix Ik1 = (∑n−1
j=1
∫ ξi+1
ξiDkj (x) dx)1≤i,j≤n−1 is the spline histopolation matrix of degree
k at the Greville points. This is an invertible matrix which defines uniquely the splinecoefficients (c1
j) characterising ϕh = Π1ϕ from the values of ϕ at the Greville points.To better understand the commuting diagram it is convenient to separate the projection
in two steps: the evaluation of the degrees to freedom σ0 and σ1 and then the interpolationof the degrees of freedom in each of the two spaces. we can express these relations in thefollowing diagram
H1(Ω) L2(Ω)
C0 C1
V0 V1
grad
σ0 σ1
G
I0 I1
grad
Π0 Π1(5.13)
We also require that this diagram commutes at all levels, which means
Gσ0 = σ1grad , grad I0 = I1G, grad Π0 = Π1grad , (5.14)
The matrix G is defined by the first relation above. Indeed, for ψ ∈ H1(Ω),
(σ1(gradψ))i =
∫ ξi
ξi−1
ψ′(x) dx = ψ(ξi)− ψ(ξi−1) = (Gσ0(ψ))i, 1 ≤ i ≤ n− 1,
so that G is the matrix of n− 1 rows and n columns
G =
−1 1 0 . . . 00 −1 1 0...
. . . . . .
0 0 −1 1
. (5.15)
Proposition 9 We have
σ1i (Π1gradψ) = σ1
i (grad Π0ψ), ∀ψ ∈ H1, ∀i, (5.16)
andΠ1gradψ = grad Π0ψ, ∀ψ ∈ H1. (5.17)
which means that the diagram (5.12) commutes.
Proof. Let us prove (5.16): Starting with ψ ∈ H1, so that gradψ ∈ L2, we have on the onehand, as ψh(ξi) = ψ(ξi) at the interpolation points
σ1i (grad Π0ψ) =
∫ ξi+1
ξi
ψ′h(x) dx = ψh(ξi+1)− ψh(ξi) = ψ(ξi+1)− ψ(ξi).
And on the other hand, as σ1i (Π1ϕ) = σ1
i (ϕ) for any ϕ ∈ L2(Ω) by definition of the histopolant
σ1i (Π1gradψ) = σ1
i (gradψ) =
∫ ξi+1
ξi
ψ′(x) dx = ψ(ξi+1)− ψ(ξi).
49
Hence we have (5.16).Finally, by construction of V0 and V1 grad Π0ψ ∈ V1 and by definition of Π1, we also
have Π1gradψ ∈ V1. Then as the degrees of freedom σ1i characterise an element of V1, the
relation (5.16) implies that Π1gradψ = grad Π0(ψ) for all ψ ∈ H1 and hence the diagram(5.12) commutes.
Application to the 1D Maxwell equations. Let us consider the 1D Maxwell equationson the interval [a, b]
∂E
∂t+∂B
∂x= −J, (5.18)
∂B
∂t+∂E
∂x= 0, (5.19)
E(a) = E(b) = 0 (5.20)
In order to apply the FEEC framework we need to choose the appropriate function spaces forE and B. One should be in H1(Ω) and the other in L2(Ω). Due to the form of the equations,both choices are possible. Let us choose here E ∈ H1
0 (Ω) and B ∈ L2(Ω). The boundaryconditions on E are in this case enforced as essential boundary conditions and put in thedefinition of the function space.
We then use the space in our commuting diagram (5.12) for the discretisation. Let usfirst apply Π1 to (5.19) using the commuting diagram property Π1grad = grad Π0:
∂Π1B
∂t+∂Π0E
∂x= 0 (5.21)
and expressing the projection with the basis functions in V1 and V0 respectively we have
Π1B =n−1∑i=1
bi(t)Dki (x), Π0E =
n−1∑i=0
ei(t)Nki (x),
and taking the derivative in x and observing that Dk0 = 0
∂Π0E
∂x=
n−1∑i=0
ei(t)(Nki )′(x) =
n−1∑i=0
ei(t)(Dki −Dk
i+1)(x) =n−1∑i=1
(ei(t)− ei−1(t))Dki (x).
Equation (5.21) being now completely expressed in the basis (Dki )i we can identify the coef-
ficients on each basis functions which yields
dbidt
+ ei(t)− ei−1(t) = 0, 1 ≤ i ≤ n− 1.
Stacking, the coefficients into vectors, b = (bi)1≤i≤n−1 and e = (ei)0≤i≤n−1 and using thematrix G defined previously in this section, this is equivalent to
db
dt+ Ge = 0. (5.22)
Remark 13 Observe that integrating (5.19) between the successive Greville points yields
d
dt
∫ ξ1
ξi−1
B(t, x) dx+ E(ξi)− E(ξi−1), 1 ≤ i ≤ n− 1
50
which can be writtend(σ1(B))
dt+ Gσ0(E) = 0. (5.23)
In order to discretise (5.18) we use the weak form. To this aim we multiply (5.18) by atest function F ∈ H1
0 and integrate (the second term by parts), which yields
d
dt
∫ b
a
E F dx−∫ b
a
B∂F
∂xdx = −
∫ b
a
J F dx ∀F ∈ H10
Now expressing E in the basis of V0 and B in the basis of V1 and taking F = Nki we find
n−1∑i=0
(dejdt
∫ b
a
Nkj N
ki dx− bj
∫ b
a
Dkj (Dk
i −Dki+1) dx+
∫ b
a
J Nki dx
)= 0 0 ≤ i ≤ n− 1
using (Nki )′ = Dk
i −Dki+1. Introducing the mass matrices in V0 and V1 respectively
M0 = (
∫ b
a
Nki N
kj dx)0≤i,j≤n−1, M1 = (
∫ b
a
Dki D
kj dx)1≤i,j≤n−1
the relations above become
M0de
dt− G>M1b = −j, (5.24)
denoting by ji =∫ baJ Nk
i dx.
5.3.3 Discrete differential forms in 3D
Let us construct spaces of differential forms whose coefficients are splines. In order to obtaina complex, we shall start by defining the space of 0-forms and then for p ≥ 1 construct thespace of p-forms by applying the exterior derivative to the (p − 1)-forms. This property isneeded to form a complex. Let us start with a finite dimensional space of 0-forms denotedby V0 constructed as a tensor product of B-Splines.
We define the 3D point x = (x1, x2, x3) and the multi-index i = (i1, i2, i3). Then the basisfunctions for 0-forms will be
Λ0i (x) = Nk
i1(x1)Nk
i2(x2)Nk
i3(x3), 1 ≤ i1 ≤ n1, 1 ≤ i2 ≤ n2, 1 ≤ i3 ≤ n3. (5.25)
And our finite dimensional space of 0-forms base on B-splines of degree k in each directionis then V0 = SpanΛ0
i i. And any discrete 0-form can be written as
ψ0h(x) =
∑i
c0i Λ
0i (x).
51
Now taking the exterior derivative of φ0h we find
dψ0h(x) =
∑i
c0i dΛ0
i (x) =∑i
c0i ((N
ki1
)′(x1)Nki2
(x2)Nki3
(x3) dx1
+Ni1(x1)(Nki2
)′(x2)Nki3
(x3) dx2 +Nki1
(x1)Nki2
(x2)(Nki3
)′(x3) dx3),
=∑i
c0i ((D
ki1−Dk
i1+1)(x1)Nki2
(x2)Nki3
(x3) dx1
+Ni1(x1)(Dki2−Dk
i2+1)(x2)Nki3
(x3) dx2 +Nki1
(x1)Nki2
(x2)(Dki3−Dk
i3+1)(x3) dx3),
=∑i
(c0i1,i2,i3
− c0i1−1,i2,i3
)Dki1
(x1)Nki2
(x2)Nki3
(x3) dx1
+ (c0i1,i2,i3
− c0i1,i2−1,i3
)Nki1
(x1)Dki2
(x2)Nki3
(x3) dx2
+ (c0i1,i2,i3
− c0i1,i2,i3−1)Nk
i1(x1)Nk
i2(x2)Dk
i3(x3) dx3,
where we used expression (5.11) for the derivative of a B-spline.In order to form a complex the finite dimensional space of 1-forms V1 should contain all
exterior derivatives of 0-forms. This will be the case if we choose as basis functions for the1-forms
Λ1i,1(x) = Dk
i1(x1)Nk
i2(x2)Nk
i3(x3) dx1 = Λ1
i,1(x) dx1,
Λ1i,2(x) = Nk
i1(x1)Dk
i2(x2)Nk
i3(x3) dx2 = Λ1
i,2(x) dx2,
Λ1i,3(x) = Nk
i1(x1)Nk
i2(x2)Dk
i3(x3) dx3 = Λ1
i,3(x) dx3.
A general finite dimensional 1-form is then represented by
f1(x) =n∑
i=1
3∑a=1
c1i,aΛ
1i,a(x) =
n∑i=1
3∑a=1
c1i,aΛ
1i,a(x) dxa.
In particular, if f1 = dφ0, there coefficients in their respective bases are related by
c1i,1 = c0
i1,i2,i3− c0
i1−1,i2,i3, c1
i,2 = c0i1,i2,i3
− c0i1,i2−1,i3
, c1i,3 = c0
i1,i2,i3− c0
i1,i2,i3−1, (5.26)
independently of the metric. It depends only on the neighboring splines.In the same way, to construct a basis for V2 we take the exterior derivative of an element
of V1. This yields
df1(x) =n∑
i=1
3∑a=1
c1i,a dΛ1
i,a(x) (5.27)
=n∑
i=1
((c1i1,i2,i3,3
− c1i1,i2−1,i3,3
)− (c1i1,i2,i3,2
− c1i1,i2,i3−1,2)
)Nki1
(x1)Dki2
(x2)Dki3
(x3) dx2 ∧ dx3 (5.28)
+((c1i1,i2,i3,1
− c1i1,i2,i3−1,1)− (c1
i1,i2,i3,3− c1
i1−1,i2,i3,3))
Dki1
(x1)Nki2
(x2)Dki3
(x3) dx3 ∧ dx1 (5.29)
+((c1i1,i2,i3,2
− c1i1−1,i2,i3,2
)− (c1i1,i2,i3,1
− c1i1,i2−1,i3,1
))
Dki1
(x1)Dki2
(x2)Nki3
(x3) dx1 ∧ dx2 (5.30)
52
In order to form a complex the space of finite dimensional 2-forms should contain allexterior derivatives of 1-forms. This will be the case if we choose as basis functions for the2-forms
Λ2i,1(x) = Nk
i1(x1)Dk
i2(x2)Dk
i3(x3) dx2 ∧ dx3 = Λ2
i,1(x) dx2 ∧ dx3,
Λ2i,2(x) = Dk
i1(x1)Nk
i2(x2)Dk
i3(x3) dx3 ∧ dx1 = Λ2
i,2(x) dx3 ∧ dx1,
Λ1i,3(x) = Dk
i1(x1)Dk
i2(x2)Nk
i3(x3) dx1 ∧ dx2 = Λ2
i,3(x) dx1 ∧ dx2.
A general discrete 2-form is then represented by
b2(x) =n∑
i=1
3∑a=1
c2i,aΛ
2i,a(x).
In particular, if b2 = df1, their coefficients in their respective bases are related by
c2i,1 = (c1
i1,i2,i3,3− c1
i1,i2−1,i3,3)− (c1
i1,i2,i3,2− c1
i1,i2,i3−1,2), (5.31)
c2i,2 = (c1
i1,i2,i3,1− c1
i1,i2,i3−1,1)− (c1i1,i2,i3,3
− c1i1−1,i2,i3,3
), (5.32)
c2i,3 = (c1
i1,i2,i3,2− c1
i1−1,i2,i3,2)− (c1
i1,i2,i3,1− c1
i1,i2−1,i3,1), (5.33)
independently of the metric. It depends only on the neighboring splines.We finally come to the last space V3 of discrete 3-forms. These are constructed such that
the exterior derivative of an element of V2 is in V3. Taking the exterior derivative of a genericelement b2 ∈ V2 yields
b2(x) =n∑
i=1
3∑a=1
c2i,a dΛ2
i,a(x)
=((c2i1,i2,i3,1
− c2i1−1,i2,i3,1
) + (c2i1,i2,i3,2
− c2i1,i2−1,i3,2
) + (c2i1,i2,i3,3
− c2i1,i2,i3−1,3)
)Dki1
(x1)Dki2
(x2)Dki3
(x3) dx1 ∧ dx2 ∧ dx3.
So the basis functions for the three forms are of the form
Λ3i (x) = Dk
i1(x1)Dk
i2(x2)Dk
i3(x3) dx1 ∧ dx2 ∧ dx3,
and a general 3-form writes
ρ3(x) =n∑
i=1
c3i Λ
3i (x). (5.34)
And the coefficients in V3 of the exterior derivative of an element of V2 are given by
c3i = (c2
i1,i2,i3,1− c2
i1−1,i2,i3,1) + (c2
i1,i2,i3,2− c2
i1,i2−1,i3,2) + (c2
i1,i2,i3,3− c2
i1,i2,i3−1,3). (5.35)
Here again this is independent of the metric an only involves neighbouring spline coefficients.
Commuting projectors The projectors are naturally based on spline integration andhistopolation at the Greville points in each direction: ξj = (ξ1j1 , ξ2j2 , ξ3j3). On a tensorproduct periodic grid, these are defined to be the grid points for odd degree splines and thegrid midpoints for even degree splines, so that the interpolation and histopolation problems
53
are always well-posed and that the tensor product spline interpolation matrix of degree p,denoted by Ip = ((Np
i1(ξ1j1)N
pi2
(ξ2j2)Npi3
(ξ3j3))i,j is square and symmetric.For any function ψ ∈ H1(Ω) we define Π0ψ =
∑i ψiΛ
0i ∈ V0, where the ψi are the unique
solution of the interpolation problem
ψ(ξj) = ψ(ξ1j1 , ξ2j2 , ξ3j3) =∑i
ψiΛ0i (xj) =
∑i1,i2,i3
ψi1,i2,i3Npi1
(ξ1j1)Npi2
(ξ2j2)Npi3
(ξ3j3).
As is usual for Kronecker products, we denote by vec(ψ(ξj)) the vector with the stackedentries (ψ(ξj), so that the interpolation problem can be written in matrix form
vec(ψ(ξj)) = Kpvec(ψi). (5.36)
The projection of a function ψ onto V0 is then uniquely defined by the coefficients ψi solutionof (5.36). As those correspond to classical Finite Element degrees of freedom, we shall denoteby σ0
i (ψ) = ψi and the vector of all degrees of freedom by
σ0(ψ) = vec(ψi) = (Kp)−1vec(ψ(ξj)). (5.37)
Now in order to get the commuting diagram grad Π0 = Π1grad , we define for any E =(E1, E2, E3)> ∈ H(curl ,Ω),
Π1E =∑i
3∑α=1
ei,αΛ1i,α ∈ V1.
On a cartesian mesh the three components decouple, and there is a separate interpolation-histopolation problem to solve for each component: the first components ei,1 are the uniquesolutions of the interpolation-histopolation problem∫ ξ1j1
ξ1j1−1
E1(x1, ξ2j2 , ξ3j3) dx1 =∑i
∫ ξ1j1
ξ1j1−1
ei,1Λ1i,1(x1, ξ2j2 , ξ3j3) dx1
=∑i1,i2,i3
ei,1
∫ ξ1j1
ξ1j1−1
Dpi1
(x1) dx1Npi2
(ξ2j2)Npi3
(ξ3j3).
We denote by Ip1,1 the corresponding tensor product interpolation-histopolation matrix
for the first component and the right hand side vec(∫ ξ1j1ξ1j1−1
E1(x1, ξ2j2 , ξ3j3) dx1) by E1(E1), E1
standing for edges in the first direction. So introducing the linear operator σ1i,α(E) = ei,α, we
have
σ11(E) = vec(ei,1) = (Ip1,1)−1vec(
∫ ξ1j1
ξ1j1−1
E1(x1, ξ2j2 , ξ3j3) dx1) = (Ip1,1)−1E1(E1). (5.38)
Similarly, the two other components of the projection Π1(E) are defined by
σ12(E) = vec(ei,2) = (Ip1,2)−1vec(
∫ ξ2j2
ξ2j2−1
E2(ξ1j1 , x2, ξ3j3) dx2) = (Ip1,2)−1E2(E2), (5.39)
54
σ13(E) = vec(ei,3) = (Ip1,3)−1vec(
∫ ξ3j3
ξ3j3−1
E3(ξ1j1 , ξ2j2 , x3) dx3) = (Ip1,3)−1E3(E3). (5.40)
We can now continue in the same way, with the definition of Π2B for B ∈ H(div ,Ω).
Π2B =∑i
3∑α=1
bi,αΛ2i,α ∈ V2.
And, introducing the linear operators σ2i,α(B) = bi,α, is 1 ≤ α ≤ 3, we have
σ21(B) = vec(bi,1) = (Ip2,1)−1vec(
∫ ξ2j2
ξ2j2−1
∫ ξ3j3
ξ3j3−1
B1(ξ1j1, x2, x3) dx2 dx3) = F1(B1), (5.41)
σ22(B) = vec(bi,2) = (Ip2,2)−1vec(
∫ ξ1j1
ξ1j1−1
∫ ξ3j3
ξ3j3−1
B2(x1, ξ2j2, x3) dx1 dx3) = F2(B2), (5.42)
σ23(B) = vec(bi,3) = (Ip2,3)−1vec(
∫ ξ1j1
ξ1j1−1
∫ ξ2j2
ξ2j2−1
B3(x1, x2, ξ3j3) dx1 dx2) = F3(B3). (5.43)
Here Fα stands for an integral on a face perpendicular the the αth direction. And finallyfor ρ ∈ L2(Ω), we can define Π3ρ =
∑i ρiΛ
3i ∈ V3 as the solution of the tensor product
histopolation problem
σ3(ρ) = vec(ρi) = (Ip3 )−1vec(
∫ ξ1j1
ξ1j1−1
∫ ξ2j2
ξ2j2−1
∫ ξ3j3
ξ3j3−1
ρ(x1, x2, x3) dx1 dx2 dx3). (5.44)
5.4 Discretisation of Maxwell’s equations with spline
differential forms
Having constructed the spline differential forms, we have already obtained relations betweenthe basis coefficients defining the exterior derivatives. All equations expressed in the strongsense (not involving the co-derivative) can be expressed using only these at the discretelevel. This can be written in a convenient matrix form by introducing the discrete gradientG defined by (5.26), the discrete curl matrix C defined by (5.31)-(5.33), and the discretedivergence D by (5.35). Note that these matrices are incidence matrices containing only 0,1, -1 coefficients.
Expanding e1h in the basis of V1 and b2
h in the basis of V2 we get
e1h(t,x) =
n∑i=1
3∑a=1
e1i,a(t)Λ
1i,a(x), b2
h(x) =n∑
j=1
3∑b=1
b2j,bΛ
2j,b(x).
Denoting respectively by e1 the vector of coefficients e1i,a(t) and by b2 the vector of
coefficients b2j,b(t), the discrete version of Faraday’s law (4.34) simply becomes
dB1
dt+ CE1 = 0,
55
and the discrete version of db2 = 0 (4.36) becomes
DB2 = 0.
Let us now consider the discretisation of the weak formulations in this setting. ForMaxwell’s equations this is needed for the Ampere equation and the Gauss law. Let usstart with the weak form of the Ampere equation. We first need to build the mass matrixcorresponding to (e1
h, f1h) for e1
h, f1h ∈ V1. Expanding e1
h, f1h in the basis of V1 we get
e1h(t,x) =
n∑i=1
3∑a=1
e1i,a(t)Λ
1i,a(x), f1
h(x) =n∑
j=1
3∑b=1
f 1j,bΛ
1j,b(x).
Using the bilinearity of the scalar product we find
(e1h, f
1h) =
n∑i=1
3∑a=1
n∑j=1
3∑b=1
e1i,a(t)f
1j,b(Λ
1i,a,Λ
1j,b). (5.45)
Then, using the definition of the scalar product of 1-forms (4.16) we get
(Λ1i,a,Λ
1j,b) =
∫Λ1
i,aΛ1i,bg
ab√g dx. (5.46)
We denote by M1 this mass matrix, stacking the multi-index (i1, i2, i3, a) into one index todefine the matrix lines, and similarly with (j1, j2, j3, b) for the columns. Denoting respectivelyby e1 the corresponding vector of coefficients e1
i,a(t) and by f1 the vector of coefficients f 1j,b(t)
we get that(e1h, f
1h) = F>1 M1E1.
Now, expanding b2h in the basis of V2, we have
b2h(t,x) =
n∑i=1
3∑a=1
b2i,a(t)Λ
2i,a(x).
On the other hand
df1h =
n∑j=1
3∑b=1
f 1j,b dΛ1
j,b(x) =n∑
j=1
3∑b=1
f 2j,bΛ
2j,b(x) = f2
h .
And the coefficients f 2j,b are related to f 1
j,b by (5.31)-(5.33). Using the discrete curl matrix C
and denoting by F2 the vector containing the coefficients f 2j,b, expression (5.31)-(5.33) can be
written in matrix formf2 = Cf1.
Now,
(b2h, df1
h) = (b2h, f
2h) =
n∑i=1
3∑a=1
n∑j=1
3∑b=1
b2i,a(t)f
2j,b(Λ
2i,a,Λ
2j,b).
56
We now need to find the expression of the scalar product of 2-forms yielding the 2-formsmass matrix. Using the formula of the scalar product (4.16) for 2-forms, with vol3 =
√g dx
together with (4.12), we get
(Λ2i,a,Λ
2j,d) =
∫Λ2
i,aΛ2j,d det(gad)
√g dx, (5.47)
where b and c are chosen such that (a, b, c) is an even permutation of (1, 2, 3) and e and f arechosen such that (d, e, f) is an even permutation of (1, 2, 3). Then det(gbcgef ) is composedof the inverse metric matrix with the line a and the column d removed. Also by checkingthe ordering for the different cases, also taking into account that the matrix G and itsinverse are symmetric, we have that det(gbcgef ) = (−1)a+dcof(G−1), where cof denotes thecofactor. Hence using the computation of the inverse of a matrix based on the adjugatematrix A−1 = (1/ detA)C>, C being the cofactor matrix, we have that
gad =1
|g|det(gbcgef ).
Whence
(Λ2i,a,Λ
2j,d) =
∫Λ2
i,aΛ2j,d
gad√g
dx. (5.48)
Let us denote by M2 this matrix. Then we can write, denoting by b2 the vector of coefficientsof b2
h
(b2h, df1
h) = (b2h, f
2h) = f>2 M2b2 = f>1 C
>M2b2.
In the FEEC setting, the mass matrices are the only matrices involving the metric thatneed to be assembled. Indeed the exterior derivative can be computed, as we saw with anexplicit formula independent of the metric.
The same procedure can then be used for the weak divergence needed in the discreteversion of Gauss’s law (4.38)
−(e1h, dψ0
h) = ψ>0 G>M1e1.
Taking the sources ρ0h in V0 and denoting by Υ0 the corresponding vector of coefficients,
and j1h ∈ V1 and denoting by j1 the corresponding vector of coefficients, we find that
(j1h, f
1h) = f>1 M1j1, (ρ0
h, ψ0h) = ψ>0 M0Υ0.
We are now ready to write the matrix form of the discrete Maxwell equations:
M1de1
dt− C>M2b2 = −M1j1, (5.49)
db2
dt+ ce1 = 0, (5.50)
G>M1e1 = M0Υ0, (5.51)
Db2 = 0. (5.52)
57
5.5 Classical Finite Element approximation of Maxwell’s
equations
In a classical derivation of a Finite Element method for Maxwell’s equations, one starts bychoosing the spaces in which the fields are defined and derive a variational formulation. ForMaxwell’s equations the natural spaces are H(curl ,Ω) for one of the fields and H(div ,Ω)for the others. As the Ampere and Faraday equations have the same structure, we have thechoice of choosing either E or B in H(curl ,Ω). Let us choose here E ∈ H(curl ,Ω) andB ∈ H(div ,Ω). This corresponds to a strong Faraday and weak Ampere formulation. Indeedmultiplying Ampere by a test function F ∈ H(curl ,Ω) and integrating by parts the curlterm and assuming that F × n vanishes on the boundary, which corresponds to essentialperfectly conducting boundary conditions.
−∫
Ω
∂E
∂t· F dx + c2
∫Ω
B · ∇ × F dx =1
ε0
∫Ω
J · F dx, ∀F ∈ H(curl ,Ω)
We notice that for ψ ∈ H1(Ω), taking F = ∇ψ ∈ H(curl ,Ω). Hence we can use it as a testfunction in the previous equation which yields
−∫
Ω
∂E
∂t· ∇ψ dx =
1
ε0
∫Ω
J · ∇ψ dx =
∫Ω
∂ρ
∂tψ dx ∀ψ ∈ H1(Ω)
using the weak form of the continuity equation∫Ω
∂ρ
∂tψ dx−
∫Ω
J · ∇ψ dx = 0 ∀ψ ∈ H1(Ω).
And integrating this equation in time between 0 and t, we find the weak Poisson equation atany time t
−∫
Ω
E · ∇ψ dx =
∫Ω
ρψ dx ∀ψ ∈ H1(Ω).
Replacing the continuous spaces by their FEEC counterparts, we have similarly
−∫
Ω
∂Eh
∂t· Fh dx + c2
∫Ω
Bh · ∇ × Fh dx =1
ε0
〈Jh,Fh〉, ∀Fh ∈ V1, (5.53)
−∫
Ω
Eh · ∇ψh dx =1
ε0
〈ρh, ψh〉, ∀ψh ∈ V0 (5.54)
For the Faraday equation, not integration by parts is necessary, so that the weak formu-lation ∫
Ω
∂B
∂t·C dx +
∫Ω
∇× E ·C dx = 0, ∀C ∈ H(div ,Ω)
is equivalent to the strong form∂B
∂t+∇× E = 0.
This property remains true at the discrete level thanks to our choice of discrete spaces.
58
5.6 Derivation of PIC model from discrete action prin-
ciple
Maxwell’s equations can be discretized in the spaces defined above keeping time continuous.The discrete scalar potential is φh ∈ V0, the discrete vector potential Ah ∈ V1, the discreteelectric field Eh ∈ V1 and the discrete magnetic field Bh ∈ V2. Because of the discrete deRham sequence satisfied by the Finite Element spaces, the fields are defined pointwise fromthe potentials
Eh = −∇φh −∂Ah
∂t, (5.55)
Bh = ∇× Ah. (5.56)
These definitions directly imply the Faraday equation by adding the curl of (5.55) and thetime derivative of (5.56)
∂Bh
∂t+∇× Eh = 0, (5.57)
and∇ · Bh = ∇ · (∇× Ah) = 0.
As in the continuous case these equations follow directly from the definition of the fields fromthe potentials.
Let us recall the continuous action principle propose by Low (2.41):
A[Xs,Vs, φ,A] =∑
s
∫fs,0(x0,v0)Ls(Xs(t; x0,v0),Vs(t; x0,v0),
dXs
dt(t; x0,v0)) dx0 dv0 dt
+ε02
∫|∇φ+
∂A
∂t|2 dx dt− 1
2µ0
∫|∇ ×A|2 dx dt. (5.58)
Now in this action principle we replace the continuous potentials by their discrete FEECapproximations, φh ∈ V0, and Ah ∈ V1. Moreover, instead of a continuum distributionfunction, we consider a discrete distribution defined a the sum of Dirac masses:
fs,h(t,x,v) =
Np,s∑k=1
wkδ(x− xk(t))δ(v − vk(t)). (5.59)
First considering only the first term of (5.58) involving f0 and the particle Lagrangian,we observe that by replacing f0, by fs,h(t = 0), we get
∑s
Np,s∑k=1
wkLs(Xs(t; xk(0),vk(0)),Vs(t; xk(0),vk(0)),dXs
dt(t; xk(0),vk(0)) dt
and as (Xs(t; xk(0),vk(0)),Vs(t; xk(0),vk(0))) is the position at time t of the macro-particlestarting at (xk(0),vk(0)), it is (xk(t),vk(t)), so that
Ls(Xs(t; xk(0),vk(0)),Vs(t; xk(0),vk(0)),dXs
dt(t; xk(0),vk(0)))
= (msvk + qsA(t,xk)) ·dxkdt−(
1
2ms|vk|2 + qsφh(t,xk)
).
59
Plugging the approximations of fh, φh and Ah into (5.58), we find
Ah[Xs,Vs, φ,Ah] =∑
s
∫fs,0(x0,v0)Ls(Xs(t; x0,v0),Vs(t; x0,v0),
dXs
dt(t; x0,v0)) dx0 dv0 dt
+ε02
∫|∇φ+
∂A
∂t|2 dx dt− 1
2µ0
∫|∇ ×A|2 dx dt. (5.60)
Finally, we separate in the above Lagrangian, the particle part, the interaction term,involving both particles and fields. This yields:
A =∑k
∫ t2
t1
(mkvk ·dxkdt− 1
2mkv
2k) dt
+ qk
∫ t2
t1
(Ah(t,xk) ·
dxkdt− φh(t,xk)
)dt
+ε02
∫|∇φh +
∂Ah
∂t|2 dx dt− 1
2µ0
∫|∇ ×Ah|2 dx dt. (5.61)
The action now depends on the dependent variables xk,vk for all k and on the discrete fieldsφh ∈ V0 and Ah ∈ V1.
The variations of A with respect to vk yield directly:
dxkdt− vk = 0.
The variations (or functional differential) of A with respect to xk yield
δA[xk; yk] = mk
∫ t2
t1
vk ·dykdt
dt
+ qk
∫ t2
t1
(yk · ∇Ah(t,xk) ·
dxkdt
+ Ah(t,xk) ·dykdt− yk · ∇φh(t,xk)
)dt
= −mk
∫ t2
t1
yk ·dvkdt
dt+ qk
∫ t2
t1
(yk · ∇Ah(t,xk) ·
dxkdt− yk · ∇φh(t,xk)
)dt
− qk∫ t2
t1
(∂Ah
∂t(t,xk) +
dxkdt· ∇Ah(t,xk)
)· yk dt
where we have integrated by parts the terms involving a time derivative in yk using that ykvanishes on the boundaries. Then as this is true for all yk and as
dxkdt· ∇Ak −∇Ah ·
dxkdt
= (∇× Ah)×dxkdt
,
we find that δA[xk; yk] = 0 yields
dvkdt
=qkmk
(−∂Ah
∂t−∇φh + vk × (∇× Ah)
)=
qkmk
(Eh + v ×Bh).
Hence we got the standard equations of motion for each particle.
60
On the other hand, the variations with respect to Ah and φh can be performed exactly asin the continuous case except that the variations are now restricted to the finite dimensionalfunction spaces V1 and V0. From these we get the weak form of Ampere’s law and Gauss’law that were already expressed in (5.53) and (5.54).
Expanding the fields Eh in the basis of V1 and Bh in the basis of V2 and denoting by eand b the respective coefficients vectors, and stacking all the particle positions in a vector Xand all the particle velocities in a vector V, we have the discrete form of the Vlasov-Maxwellsystem on the left and to see the correspondence the continuous form on the right:
X = V x = v,
V = M−1p Mq
(1(X)e + B(X,b)V
)v =
qsms
(E + v ×B),
e = M−11
(C>M2b(t)− 1(X)>MqV
) ∂E
∂t= ∇× B− J,
b = −Ce(t)∂B
∂t= −∇× E.
where we introduced the notations B(X,b) to represent the discrete v × B, and 1(X) forthe matrix containing all the basis functions in V1 evaluated at all the particle positions.Moreover Mp is the diagonal matrix containing the mass mk of all the particles and Mq is thediagonal matrix containing the charge qk of all the particles.
It is possible to check that this defines a non canonical Hamiltonian system with hamil-tonian
H =1
2V>MpV +
1
2e>M1e +
1
2b>M2b
such that∇H = (0,MpV,M1e,M2b)>
and with the Poisson matrix
J =
0 M−1
p 0 0−M−1
p B(X,b) 1(X)M−11 0
0 −M−11 1(X)> 0 M−1
1 C>
0 0 −CM−11 0
.
61
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