LEAST-SQUARES FINITE ELEMENT METHODS

Max Gunzburger

Department of Scientific Computing

Florida State University

gunzburg@fsu.edu

WHY STUDY LEAST-SQUARES FINITE ELEMENT METHODS?

• Finite element methods were first developed and analyzed in the Rayleigh-Ritzsetting, i.e., problems whose solutions can be characterized as minimizers ofconvex, quadratic functionals

– examples include

- the equations of linear elasticity whose solutions can be characterized

as minimizers of the quadratic potential energy functional

- the Poisson problem

−∆φ = f in Ω and φ = 0 on Γ = ∂Ω

whose solution φ can be characterized as the minimizer, over a

suitable class of functions, of the Dirichlet functional1

2

∫

Ω

|∇φ|2 dΩ −

∫

Ω

fφ dΩ

• The Rayleigh-Ritz setting results in finite element methods having the follow-ing desirable features

1. general regions and boundary conditions are relatively easy to handle andhigher-order accuracy is relatively easy to achieve

2. the conformity of the finite element spaces is sufficient to guarantee stabilityand optimal accuracy

3. for systems of PDEs, all variables can be approximated using a single typeof finite element space

4. the resulting linear systems are

a) sparse; b) symmetric; c) positive definite

• FEMs were quickly applied in other settings

– motivated by the fact that properties 1 and 4a are retained by all FEMs

• Mixed FEMs arose from minimization problems constrained by PDEs

– result in indefinite problems

– the only other property (sometimes) retained from the Rayleigh-Ritz settingis symmetry

– onerous compatibility conditions between finite element spaces often arise

• More generally, Galerkin FEMs are defined by forcing the residual of the PDEto be “orthogonal” to the finite element subspace

– other than 1 and 4a, none of the advantages of the Rayleigh-Ritz settingare retained

• It is a testament to the importance of advantage 1 that despite the loss ofother advantages, mixed and Galerkin FEMs are in widespread use

• Not surprisingly,

despite the success of mixed and Galerkin FEMs,

there has been substantial interest and effort devoted to

developing finite element approaches that recover at least some of the

advantages of the Rayleigh-Ritz setting

– stabilized FEMs

– penalty methods

• Least-squares finite element methods can be viewed as another attempt atretaining the advantages of the Rayleigh-Ritz setting even for much moregeneral problems

– in fact, they offer the possibility of, in principle, retaining all of the advan-tages of that setting for practically any PDE problem

– let’s see how this is done

Straightforward LSFEM

• Consider the problem

Lu = f in Ω and Ru = g on Γ

– here Lu = f is a PDE and Ru = g is a boundary condition

• We assume nothing about this problem other than it is well posed

– there exists

a solution Hilbert space S

data Hilbert spaces HΩ and HΓ

positive constants α1 and α2

such that the PDE estimate

α1‖u‖2S ≤ ‖Lu‖2

HΩ+ ‖Ru‖2

HΓ≤ α2‖u‖

2S ∀u ∈ S

holds

• Now, consider the least-squares functional

J(u; f, g) = ‖Lu− f‖2HΩ

+ ‖Ru− g‖2HΓ

and the unconstrained minimization problem

minu∈S

J(u; f, g)

so that

(Lv,Lu)HΩ+ (Rv,Ru)HΓ

= (Lv, f)HΩ+ (Rv, g)HΓ

∀ v ∈ S

• A LSFEM can be defined by

– choosing a finite element subspace Sh ⊂ S

– then restricting the minimization problem to the subspace

• Thus, the LSFEM approximation uh ∈ Sh is the solution of the problem

minuh∈Sh

J(uh; f, g)

• All the desirable properties of the Rayleigh-Ritz setting are recovered!

• The key is the norm equivalence of the functional

α1‖u‖2S ≤ J(u; 0, 0) ≤ α2‖u‖

2S ∀u ∈ S

which follows from

J(u; 0, 0) = ‖Lu‖2HΩ

+ ‖Ru‖2HΓ

∀u ∈ S

and the well-posedness estimate

α1‖u‖2S ≤ ‖Lu‖2

HΩ+ ‖Ru‖2

HΓ≤ α2‖u‖

2S ∀u ∈ S

• Unfortunately, this is not the whole story

– this straightforward procedure does not always lead to a practical method

Practicality issues

• We refer to a finite element method as being practical if it meets the followingcriteria

– bases for the finite element spaces are easily constructed

– linear systems are easily assembled

– linear systems are relatively well conditioned

• In judging whether or not a LSFEM is practical, we will measure it up againstGalerkin FEMs for the Poisson equation

• In particular, we will ask the questions:

– can we use standard, piecewise polynomial spaces that are merely continu-ous and for which bases are easily constructed?

– can the assembly of the linear systems be accomplished by merely applyingquadrature rules to integrals?

– are the condition number of the linear systems of O(h−2)?

• Even in the Raleigh-Ritz setting, not all finite element methods are practical

– for example, consider the following functional1

2

∫

Ω

ψψ dΩ +

∫

Ω

fψ dΩ

whose minimizers over a suitable function space are weak solutions of theDirichlet problem for the biharmonic equation:

2ψ = f in Ω and ψ = 0 and∂ψ

∂n= 0 on Γ

– the Rayleigh-Ritz finite element setting

- requires the use of C1(Ω) finite element spaces

- results in difficult assemblies of linear sysems

- resulting condition numbers are of O(h−4)

– none of the three practicality criteria are met

• We will later consider the keys to the practicality of LSFEMs

Further observations about the straightforward LSFEM

• We began with

– the PDE problem

Lu = f in Ω and Ru = g on Γ

– the related PDE estimate

α1‖u‖2S ≤ ‖Lu‖2

HΩ+ ‖Ru‖2

HΓ≤ α2‖u‖

2S ∀u ∈ S

– the least-squares functional

J(u; f, g) = ‖Lu− f‖2HΩ

+ ‖Ru− g‖2HΓ

– and the unconstrained minimization problem

minu∈S

J(u; f, g)

• A least-squares functional may be viewed as an “artificial” energy that playsthe same role for LSFEMs as a bona fide physically energy plays for Rayleigh-Ritz FEMs

• The least-squares functional J(·; ·, ·) measures the residuals of the PDE andboundary condition using the data space norms HΩ and HΓ, respectively

• The minimization problem seeks a solution in the solution space S for whichPDE estimate is satisfied

• The PDE system and the minimization problem are equivalent in the sensethat u ∈ S is a solution of the latter if and only if it is also a solution, perhapsin a generalized sense, of the former

• A LSFEM was defined by

– choosing a family of finite element subspaces Sh ⊂ S parameterized by htending to zero

– then restricting the minimization problem to the subspaces

so that the LSFEM approximation uh ∈ Sh to the solution u ∈ S of theleast-squares minimization problem or of the PDE problem is the solution ofthe discrete least-squares minimization problem

minuh∈Sh

J(uh; f, g)

• The Euler-Lagrange equations corresponding to the two minimization prob-lems are given by

seek u ∈ S such that B(u, v) = F (v) ∀ v ∈ S

seek uh ∈ Sh such that B(uh, vh) = F (vh) ∀ vh ∈ Sh

respectively, where we have the bilinear form

B(u, v) = (Lv,Lu)HΩ+ (Rv,Ru)HΓ

∀u, v ∈ S

and the linear functional

F (v) = (Lv, f)HΩ+ (Rv, g)HΓ

∀ v ∈ S

• The norm-equivalence of the functional now takes the form

α1‖u‖2S = J(u; 0, 0) = B(u, u) ≤ α2‖u‖

2S

• If we choose a basis UjJj=1, where J = dim(Sh), then we have that

uh =J∑

j=1

cjUj

for some constants cjJj=1

• Then the discretized problem is equivalent to the linear system

K c = f

where we have that

cj = cj for j = 1, . . . , J

Kij = B(Ui, Uj) = (LUi,LUj)HΩ+ (RUi,RUj)HΓ

for i, j = 1, . . . , J

fi = F (Ui) = (LUi, f)HΩ+ (RUi, g)HΓ

for i = 1, . . . , J

• The results of the following theorem follow directly from the PDE estimateand the resulting norm equivalence of the least-squares functional

THEOREM - Assume that the PDE estimate holds and that Sh ⊂ S. Then,

– the bilinear form B(·, ·) is continuous, symmetric, and coercive and the linearfunctional F (·) is continuous

– the problem B(u, v) = F (v) for all v ∈ S has a unique solution u ∈ S thatis also the unique solution of the least-squares minimization problem

– the problem B(uh, vh) = F (vh) for all vh ∈ Sh has a unique solution uh ∈ Sh

that is also the unique solution of the discrete least-squares minimizationproblem

– for some constant C > 0, we have that

‖u‖S ≤ C(‖f‖HΩ+ ‖g‖HΓ

) and ‖uh‖S ≤ C(‖f‖HΩ+ ‖g‖HΓ

)

– for some constant C > 0, u and uh satisfy the optimal error estimate

‖u− uh‖S ≤ C infvh∈Sh

‖u− vh‖S

– the matrix K is symmetric and positive definite

• Note that it is not assumed that the PDE problem is self-adjoint or positiveas it would have to be in the Rayleigh-Ritz setting

– it is only assumed that it is well posed

• Despite the generality of the PDE problem, the LSFEM based recovers all thedesirable features of FEMs in the Rayleigh-Ritz setting

• The least-squares finite element approximation is optimally accurate with re-spect to solution norm ‖ · ‖S for which the PDE problem is well posed

• In defining the least-squares minization principle, one need not restrict thespaces S and Sh to satisfy the boundary conditions

– instead, any boundary conditions can be imposed weakly by including theresidual Ru− g of the boundary condition in the least-squares functionalJ(·; ·, ·)

– thus, we see that LSFEMs possess a desirable feature that is absent evenfrom standard FEMs in the Rayleigh-Ritz setting

- the imposition of boundary conditions can be effected through the

functional and need not be imposed on the finite element spaces

– notwithstanding this advantage, one can, if one wishes, impose essentialboundary conditions on the space S

- in this case, all terms involving the boundary condition are omitted and

we also set HΩ = H

• Note also that since

J(uh; f, g) = ‖Luh − f‖2HΩ

+ ‖Ruh − g‖2HΓ

= B(uh, uh) − 2F (uh) + (f, f)HΩ+ (g, g)HΓ

,

the least-square functional J(uh; f, g) provides a computable indicator for theresidual error in the LSFEM approximation uh

– such indicators are in widespread used for grid adaption

• The problemB(u, v) = F (v) ∀v ∈ S

displays the normal equation form typical of least-squares systems, e.g., recallthat we have that

B(u, v) = (Lv,Lu)HΩ+ (Rv,Ru)HΓ

– it is important to note that since L is a differential operator, this probleminvolves the higher-order differential operator L∗L

– we shall see that this observation has a profound effect on how practicalLSFEMs are defined

• The complete recovery, in general settings, of all desirable features of theRayleigh-Ritz setting is what makes LSFEMs intriguing and attractive

– but, what about the practicality of the straightforward LSFEM?

– we explore this issue using examples

An impractical application of the straightforward LSFEM

• Consider the problem

−∆u = f in Ω and u = 0 on Γ

– of course, this is a problem which fits into the Rayleigh-Ritz framework sothat there is no apparent need to use any other type of FEM

- inhomogeneous Dirichlet boundary conditions provide a situation in

which one might want to use LSFEMs even for this problem

– however, let us use the LSFEM method anyway, and see what happens

• We have (under some assumptions on the domain Ω) that the PDE estimateholds with

S = H2(Ω) ∩H10(Ω) H = L2(Ω) L = −∆

– note that, for simplicity, we impose the boundary condition on the solutionsspace S

• We then have that, for all u, v ∈ H2(Ω) ∩H10(Ω),

J(u; f) = ‖∆u + f‖20 F (v) =

∫

Ω

f∆v dΩ B(u, v) =

∫

Ω

∆v∆u dΩ

• Note that minimizing the least-squares functional has turned the second-orderPoisson problem into a fourth-order problem

• A LSFEM is defined by choosing a subspace Sh ⊂ S = H2(Ω) ∩H10(Ω) and

then minimizing the functional J(·; f) over Sh

• It is well known that in this case, the finite element space Sh has to consistof continuously differentiable functions

– this requirement greatly complicates the construction of bases and the as-sembly of the matrix problem

• Furthermore, it is also well known that the condition number of the matrixproblem is O(h−4)

– this should be contrasted with the O(h−2) condition number obtainedthrough a Rayleigh-Ritz discretization of the Poisson equation

• Thus, for this problem, the straightforward LSFEM fails all three practicalitytests

• It is also true that PDE estimate holds with

S = H10(Ω) H = H−1(Ω)

– one could then develop a LSFEM based on the functional

J(u; f) = ‖∆u + f‖2−1

and the solution spaceS = H1

0(Ω)

• This approach would allow one to use a finite element space Sh consisting ofmerely continuous functions so that bases may be easily constructed

• Moreover, it can be shown that because of the use of the H−1(Ω) norm inthe functional, the condition number of the resulting matrix system is O(h−2)which is the same as for a Rayleigh-Ritz discretization

• However, the H−1(Ω) inner product is computed by inverting the Laplacianoperator which makes the assembly of the matrix problem difficult

– it also leads to the loss of property 4a, the sparsity of the matrix K

• So, as it stands, the straightforward LSFEM remains impractical for thesecond-order Poisson problem.

A practical application of the straightforward LSFEM

• Consider the problem (say in two dimensions)

−∇ · u = f and ∇× u = g in Ω and n · u = 0 on Γ

• Hereu ∈ S = H1

n(Ω) = v ∈ H1(Ω) | n · v = 0 on Γ

andf, g ∈ H = L2

0(Ω) × L2s(Ω)

where

L20(Ω) = f ∈ L2(Ω) |

∫

Ω

f dΩ = 0

andL2s(Ω) = g ∈ L2(Ω) | ∇ · g = 0 in Ω

• We then have that the PDE estimate holds so that we may define the least-squares functional

J(u; f, g) = ‖∇ · u + f‖20 + ‖∇ × u− g‖2

0 ∀u ∈ S = H1n(Ω)

– we then have that

B(u,v) =

∫

Ω

((∇·u)(∇·v)+(∇×u)·(∇×v)

)dΩ ∀u,v ∈ S = H1

n(Ω)

and

F (v) =

∫

Ω

(− f∇ · v + g · ∇ × v

)dΩ ∀v ∈ S = H1

n(Ω)

• A LSFEM is defined by choosing a subspace Sh ⊂ S = H1n(Ω) and then

minimizing J(·; f ,g) over Sh

• In this case, the LSFEM recovers all the good properties of the Rayleigh-Ritzsetting and also satisfies all three practicality criteria

– since we merely require that Sh ⊂ H1n(Ω), we can choose standard finite

element spaces for which bases are easily constructed

– furthermore, since the functional J(·; ·, ·) only involves L2(Ω) inner prod-ucts, the assembly of the matrix system is accomplished in a standardmanner

– finally, it can be shown that the condition number of the matrix system isO(h−2).

Norm-equivalence vs. practicality

• We have referred to a functional satisfying

α1‖u‖2S ≤ J(u; 0, 0) ≤ α2‖u‖

2S

as being norm equivalent

– this property causes the LSFEM defined by minimizing such a functional torecover all the desirable properties of the Rayleigh-Ritz setting

• However, the norms that enter the definition of the functional J(·; ·, ·) as wellas the form of the PDE system can render the resulting LSFEM impractical

• Thus, in order to define a practical LSFEM, one may have to define a least-squares functional that is not norm equivalent

– we take up this issue in later

– here, we examine the examples just discussed to see what guidance theygive us about what makes a LSFEM practical

First-order system form of the PDE

• Perhaps the most important observation that can be made from the examplesis that the div-curl example involved

– a first-order system of PDEs

- so that the normal equation operator L∗L is merely a second-order

differential operator

– a LSFEM that allowed for the easy construction of finite element bases

- because one could work with merely continuous finite element spaces

– and resulted in a matrix systems with relative good O(h−2) conditioning

• As a result, modern LSFEMs are based on first-order formulations of PDEsystems

– of course, many if not most PDEs of practical interest are not usually posedas first-order systems

• Thus, the first step in defining a LSFEM should be recasting a given PDEsystem into a first-order system

• Unfortunately, there is no unique way to do this

– for example, the three problems

u + ∇φ = 0 in Ω∇ · u = f in Ωφ = 0 on Γ

u + ∇φ = 0 in Ω∇ · u = f in Ω∇× u = 0 in Ωφ = 0 on Γ

∇ · u = f in Ω∇× u = 0 in Ωn × u = 0 on Γ

are all first-order systems that are equivalent to the Poisson problem

−∆φ = f in Ω φ = 0 on Γ

– each happens to be norm equivalent, but with respect to different norms

– if we assume that in each case the boundary condition is imposed on thesolutions space, we have that the space S is respectively given by

H10(Ω) ×H(Ω, div) H1

0(Ω) × H1(Ω) H1τ(Ω)

whereH(Ω, div) = v ∈ L2(Ω) |∇ · v ∈ L2(Ω)

H1τ(Ω) = v ∈ H1(Ω) |n × v = 0 on Γ

Functionals formed using L2 norms of equation residuals

• Another observation that can be gleaned from the examples is that if onewants to be able to assemble the matrix system using standard finite ele-ment techniques, then one should use L2 norms of equation residuals in thedefinition of the least-squares functional

– unfortunately, it is not always the case that the resulting least-squares func-tional is norm equivalent

– let us explore this issue in more detail

• Consider the Stokes problem

−∆u + ∇p = f , ∇ · u = 0 in Ω and u = 0 on Γ

• The most popular LSFEM for this problem is based on the first-order system

∇×ω+∇p = f , ω = ∇×u, ∇·u = 0 in Ω and u = 0 on Γ

that is known for obvious reasons as the velocity-vorticity-pressure formulation

• One would then be tempted to use the functional

J0(u,ω, p; f) = ‖∇ × ω + ∇p− f‖20 + ‖∇ × u − ω‖2

0 + ‖∇ · u‖20

that involves only L2(Ω) norms of equation residuals

– indeed, this is the most popular approach for defining LSFEM for the Stokesequations

– unfortunately, this functional is not norm equivalent

• On the other hand, the functional

J−1(u,ω, p; f) = ‖∇ × ω + ∇p− f‖2−1 + ‖∇ × u − ω‖2

0 + ‖∇ · u‖20

is norm equivalent to‖u‖2

1 + ‖ω‖20 + ‖p‖2

0

but involves the impractical H−1(Ω) norm

• So, on the one hand

– the lack of norm equivalence for the functional J0(·, ·, ·; ·) results in a lossof accuracy of the LSFEM approximations based on that functional

on the other hand

– the appearance of the H−1(Ω) norm in the functional J−1(·, ·, ·; ·) results inan impractical LSFEM because the matrix systems are not easily assembled

• To resolve this dilemma, we have to turn to more sophisticated LSFEMs

– here, we only give a brief and superficial review of this subject

for details see the Springer book with Pavel Bochev

MORE SOPHISTICATED LSFEMS

• We refer to the variational principle

minu∈S

J(u; f, g)

as the continuous least-squares principle

• Instead of using this principle as the basis for defining a LSFEM, we nowchoose

– a finite element space Sh and

– a convex quadratic functional Jh(·; f, g) defined over Sh

• The pair Sh, Jh(·; f, g) gives rise to the discrete least-squares principle

minuh∈Sh

Jh(uh; f, g)

• Since we only require that the functional Jh(·; f, g) be defined for functionsin Sh

and not necessarily over S,

we refer to LSFEMs constructed in this manner as discrete LSFEMs

• The functional Jh(·; f, g) is required to satisfy the following non-restrictiveassumptions

H1. There exists

a discrete energy inner product (·, ·)h : Sh × Sh 7→ ℜ

and

a discrete energy norm ‖ · ‖h = (·, ·)1/2h

such that

Jh(uh; 0, 0) = (uh, uh)h = ‖uh‖2

h for all uh ∈ Sh

H2. There exist bilinear forms E(·, ·) and T (·, ·) such that for all smoothfunctions u ∈ S and all uh ∈ Sh

Jh(uh;Lu,Ru) = ‖u− uh‖2

h + E(u, uh) + T (u, u)

• The two assumptions are sufficient to prove the following results

THEOREM - Assume that

the hypotheses H1 and H2 hold for Sh, Jh(·; f, g)

and

let u denote a sufficiently smooth solution of PDE problem

Then,

the discrete least-squares principle has a unique solution uh ∈ Sh

Moreover, uh satisfies

‖u− uh‖h ≤ infvh∈Sh

‖u− vh‖h + supvh∈Sh

E(u, vh)

‖vh‖h

• A discrete least-squares functional Jh(·; f, g) is referred to as being order r-consistent if, for any sufficiently smooth u ∈ S,

E(u, uh) ≤ C(u)hr for some r > 0

• If Jh(·; f, g) is order r-consistent, then we have the error estimate

‖u− uh‖h ≤ infvh∈Sh

‖u− vh‖h + C(u)hr

• Defining pairs Sh, Jh(·; f, g) such that the assumptions H1 and H2 aresatisfied is not a difficult task

Constructing discrete least-squares functionals

• We have an estimate for the error with respect to the discrete norm ‖ · ‖h

– of greater interest is estimating errors using the (mesh-independent) solu-tion norm ‖ · ‖S

- since Sh ⊂ S, it is true that ‖ · ‖S acts as another norm on Sh

- since Sh is finite dimensional, the two norms ‖ · ‖h and ‖ · ‖S are

comparable

– however, the comparability constants may depend on h

- if they do, then error estimates in the norm ‖ · ‖S will surely involve

constants that depend on inverse powers of h

- at the least, accuracy may be compromised

• We conclude that the hypotheses H1 and H2 do not sufficiently connectJh(·; f, g) to the PDE problem for one to determine much about the propertiesof the error estimate with respect to ‖ · ‖S norm

– thus, we now discuss how to construct discrete least-squares functionals sothat we can get a handle on these properties

• We assume that we have the PDE estimate and the norm-equivalence of thefunctional J(·; f, g)

– these are with respect to the norms associated with the solution space Sand the data spaces HΩ and HΓ

• LetDS DΩ DΓ

denote norm-generating operators that allow us to relate the norms on

S HΩ HΓ

respectively, to L2(Ω) norms

– i.e., we have that, for all u ∈ S, f ∈ HΩ, and g ∈ HΓ,

‖u‖S = ‖DSu‖0 ‖f‖HΩ= ‖DΩf‖0 ‖g‖HΓ

= ‖DΓg‖0,Γ

• We then let

Jh(uh; f, g) = ‖Dh

Ω(Lhuh −QhΩf)‖2

0 + ‖DhΓ(Rhuh −Qh

Γg)‖20

whereDh

Ω DhΓ Lh Rh

are approximations of the corresponding operators and

QhΩ : HΩ 7→ L2(Ω) Qh

Γ : HΓ 7→ L2(Γ)

are projections

• It can be shown that Jh(uh; f, g) satisfies hypothesis H2 with a specific form

for the truncation term E(u, vh)

• The main objective in choosing Lh and Rh is to make Jh(u; f, g) as small aspossible for the exact solution u of the PDE problem

– an appropriate choice is to choose discrete operators that will lead to trun-cation errors of order r

• On the other hand, DΩ and DΓ define the “energy” balance of for the PDEproblem that is not reflected by the L2 norms

– i.e., the proper scaling between data and solution spaces

– as a result, the main objective in the choice of DhΩ and Dh

Γ is to ensurethat the scaling induced by Jh(·; f, g) is as close as possible to the PDEestimate

- i.e., to “bind” Sh, Jh(·; f, g) to the energy balance of S, J(·; f, g)

• For norm-equivalent discrete least-squares principles, Jh(·, f, g) satisfies

α1‖uh‖S ≤ Jh(·; 0, 0) ≤ α2‖u

h‖S ∀uh ∈ Sh

– if the finite element space satisfies standard inverse assumptions, minimizersof this functional satisfy the error estimate

‖u−uh‖S ≤ C

infvh∈Sh

‖u−vh‖S +(

infvh∈Sh

‖u−vh‖h+ supvh∈Sh

E(u, vh)

‖vh‖h

)

• For quasi norm-equivalent discrete least-squares principles, Jh(·; f, g) satisfies

αh1‖uh‖S ≤ Jh(·; 0, 0) ≤ αh2‖u

h‖S ,

where αh1 > 0 and αh2 > 0 for all h > 0 but may depend on h.

– under additional assumptions, error estimates can also be derived in thiscase

COMPATIBLE LSFEMS

• The class of methods known as compatible LSFMEs is motivated, amongother things, by the desire to achieve good conservation properties

– e.g., to achieve as well as possible mass conservation

∇ · u = 0 in Ω

• Mass conservation can be achieved

– pointwise: at every point

– locally: in the sense that

∫

element

∇ · u dΩ = 0

– globally: in the sense that

∫

Ω

∇ · u dΩ = 0

• It is usually desired that mass conservation be achieved at least locally

– this is one motivation for using the first-order system form of the Poissonequation

• Consider the problem

u + ∇φ = 0 and ∇ · u = 0 in Ω u · n = 0 on Γ

– this is a first-order system form of a Poisson problem

• There are two mixed finite element methods for this system

– they are based on the classical Dirichlet and Kelvin principles

– the require that the finite element spaces used for the approximation of thevector u and the scalar φ satisfy a compatibility condition

- the conformity of the spaces is not sufficient to guarantee stability

• In particular, standard nodal finite element spaces cannot be used for bothvariables

– such a space can be used for the approximation of the scalar variable if oneuses the Dirichlet principle

- then the vector variable approximation has to belong to the image of

the scalar space under the gradient operator

- the space for the vector is not a standard finite element space

– such a space cannot be used for the approximation of the vector variable,even if one uses the Kelvin principle

- one has to use more complex, non-standard finite element spaces

• This is one reason advanced for using LSFEMs for this problem

– one can use the straightforward LSFEM with standard continuous, nodalfinite element spaces for both variables

– one obtains optimally accurate approximations with respect to the normsin which the problem is well posed

but

– one does not obtain optimal accuracy for the vector variable approximationwith respect to the L2(Ω) norm

– one does not do well at mass conservation

• However, if

– one uses the approximation space used in the Dirichlet principle-based mixedmethod for the scalar variable

and if

– one uses the approximation space used in the Kelvin principle-based mixedmethod for the vector variable

then one obtains

– optimal accuracy for all variables with respect to L2(Ω) norms

– the same excellent local mass conservation as is observed in mixed methodsbased on the Kelvin principle

• We refer to such a LSFEMs as a compatible LSFEM because it is “compatible”with mixed methods for the same problem

• Thus, by using spaces that are stable for mixed methods, one can define aLSFEM that has excellent mass conservation properties

• This means that to get excellent mass conservation, one has to give up oneof the advantageous features of the Rayleigh-Ritz setting

– one cannot use the same standard finite element spaces for all variables

• However, compatible LSFEMs retain all of the other advantages of the Rayleigh-Ritz setting, including the positive definiteness of the discrete system and theavoidance of onerous compatibility conditions

– mixed finite element methods do not possess these advantages

• Compatible LSFEMs have also been developed for other PDEs such as curl-curlsystems arising in electromagnetics

WHAT IS THE STATUS OF LSFEMS TODAY?

• Least-squares finite element methods have been developed and analyzed for avariety of problems

• The titles of the chapters of our forthcoming book give an indication of thescope of research into LSFEMs

– Classical variational methods

– Alternative variational formulations

– Mathematical foundations of least-squares finite element methods

– First-order Agmon-Douglis-Nirenberg systems

– Basic first-order systems

– Application to key elliptic problems

– The Navier-Stokes equations

– Dissipative time dependent problems

– Hyperbolic problems

– Control and optimization problems

– Other topics that were not discussed elsewhere

SOME OPEN PROBLEMS IN LSFEMS

• We close with a brief discussion of some of the open problems that exist inthe theory and application of LSFEMs

Hyperbolic PDEs

• The recovery of the Rayleigh-Ritz properties by LSFEMs relies on the existenceof Hilbert spaces that validate the PDE estimate

– such bounds are natural for elliptic PDEs and can be derived for any suchPDE by using the Agmon-Douglis-Nirenberg theory

• On the other hand, for hyperbolic PDEs, such bounds are not so natural,partly because they admit data in Lp spaces and their solutions may havecontact discontinuities and shock waves

• Recall that the weak form of the least-squares principle can be viewed as aGalerkin method applied to a higher-order PDE

– as a result, LSFEMs for hyperbolic equations designed using a Hilbert spacesetting are equivalent to a Galerkin discretization of a degenerate ellipticPDE

– the result is a LSFEM that will have excellent stability properties but willsmear shocks and discontinuities

• To illustrate some of the pitfalls that can be encountered with hyperbolicPDEs, it suffices to consider the simple linear convection-reaction problem

∇ · (bu) + cu = f in Ω and u = g on Γ−

whereb(x) is a given “velocity” vectorc(x) is a given functionΓ− = x ∈ Γ |n(x) · b(x) < 0 is the inflow part of the boundary Γ

• A straightforward L2(Ω) norm-based least-squares principle is defined by min-imizing the functional

J(v; f, g) = ‖∇ · (bv) + cv − f‖20 + ‖v − g‖2

0,Γ−

over the Hilbert space

S = u ∈ L2(Ω) | Lu = ∇ · (bv) + cv ∈ L2(Ω)

• The following result was obtained in [Bochev & Choi]

THEOREM - Assume that Γ− is non-characteristic and c + 12∇ · b ≥ σ > 0 for

some constant σ

Then,

J(v; 0, 0) is norm equivalent to the graph norm ‖v‖2S = ‖v‖2

0 + ‖Lv‖20

for every f ∈ L2(Ω) and g ∈ L2(Γ−), J(v; f, g) has a unique minimizer u ∈ S

for that u we have that J(u; 0, 0) ≤ ‖f‖0 + ‖g‖20,Γ−

• This theorem shows that if the data belongs to L2, all the prerequisites neededto define a LSFEM are fulfilled

– we can proceed as before to define a LSFEM in a straightforward way

• However, the convection-reaction problem is meaningful even if the data fbelongs only to L1(Ω)

– in this case, proper solution and data spaces are given by S = v ∈L1(Ω) | ∇ · (bv) ∈ L1(Ω) and H = L1(Ω), respectively

– it has been shown [Guermond] that L is an isomorphism S 7→ H and sowe have the bound α1‖u‖S ≤ ‖Lu‖H ∀u ∈ S

• Now, consider the unconstrained minimization problem associated with thespaces S and H:

minu∈S

J1(u; f), where J1(u; f) = ‖Lu− f‖L1(Ω) =

∫

Ω

|Lu− f | dΩ

• For our model problem, this is the “correct” minimization problem that, whenrestricted to Sh ⊂ S, will have solutions that do not smear discontinuities

– this fact has been recognized independently in [Jiang] and more recently in[Guermond]

• In [Guermond], it is also shown that under some reasonable assumptions onSh, the discrete problem

minuh∈Sh

J1(uh; f)

has at least one global minimizer, no local minimizers, and a solution thatsatisfies the stability bound ‖uh‖S ≤ C‖f‖H

• We can view this setting as yet another example of the conflict betweenpracticality and optimality

– in this case, however, the practicality issue is much more severe because

- the least-squares functional J1(·; f) is not differentiable

- we cannot write a first-order optimality condition

- the discrete problem does not give rise to a matrix problem

– this is the chief reason that so far there are only two examples [Jiang,Guermond] of FEMs for the model problem based on the L1 optimizationproblem

– in [Guermond], the minimizer is approximated by solving a sequence ofregularized L1 optimization problems that are differentiable

– [Jiang] uses a sequence of more conventional L2 least-squares approaches,but defined using an adaptively weighted L2 inner product

- the weights are used to weaken contributions to the least-squares

functional from elements that contain solution discontinuities

• At this point, there is very limited experience with solving hyperbolic PDEsby minimizing functionals over Banach spaces

– for problems with non-smooth data, computational experiments with themethods of [Guermond] and [Jiang] show that they are superior to LSFEMsdefined through the minimization of L2-based functionals

– most notable is their ability to provide sharp discontinuity profiles withoutover- and under-shooting

– a series of experiments in [Guermond] also points strongly towards a pos-sibility that the numerical solutions actually obey a maximum principle ongeneral unstructured grids and that the L1-based algorithm seems to beable to select viscosity solutions

- however, at present, there are no mathematical confirmations of these

facts

- nor is it known whether such algorithms for hyperbolic conservation

laws are able to provide accurate shock positions and speeds

• Despite the promise of L1 optimization techniques, the state of LSFEMs forhyperbolic problems is far from satisfactory

– straightforward L2 norm-based LSFEMs are clearly not the most appropriateas they are based on the “wrong” stability estimate for the problem

– L1 norm-based techniques give far better results but are more complex and,in some cases require the solution of nonlinear optimization problems

• Thus, the jury is still out on whether or not it is possible to define a simple,robust, and efficient LSFEMs for hyperbolic problems that will be competitivewith specially designed, upwind schemes employing flux limiters

Mass conservation

• LSFEMs for the Poisson equation can be implemented so that the approxima-tion locally conserves mass

• Achieving local mass conservation in LSFEMs for incompressible, viscous flowsremains an important open problem

• All existing LSFEMs for incompressible, viscous flows conserve mass only ap-proximately so that

‖∇ · uh‖0 = O(hr)

where r is the approximation order of the finite element space

• For low-order elements, which are among the most popular and easy to use,LSFEMs have experienced severe problems with mass conservation

• For LSFEMs based on the velocity-vorticity-pressure system, these problemswere first identified in [Chang & Nelson] where also a solution was proposedthat combines least-squares principles and Lagrange multipliers to achieveelement-wise mass conservation

– the method achieves remarkable local conservation but compromises themain motivation underlying LSFEMs: to recover a Rayleigh-Ritz setting forthe PDE

- in particular, positive definiteness is lost

• An alternative to exact local mass conservation is an LSFEM with enhancedtotal mass conservation

– this can be effected by increasing the importance of the continuity residualby using weights

– for example, a weighted LSFEM for the velocity-vorticity-pressure formula-tion of the Stokes problem using the functional

JW (ω, p,u) = ‖∇×ω+∇p−f‖20+

∑

K∈Th

h2K

(W‖∇·u‖2

0,K+‖∇×u−ω‖20,K

)

was studied in [Deang & G.] where numerical studies showed that fairlya small weight, e.g., W = 10, helps to significantly improve total massconservation

• Thus, for the Stokes problem, at present there are methods that either

– recover local mass conservation but forfeit some important advantages ofthe Rayleigh-Ritz settings

or

– retain all those advantages but can at best provide improved global conser-vation

• It is of interest to explore whether or not it is possible to develop LSFEMsfor viscous flows that retain all the Rayleigh-Ritz advantages and at the sametime locally conserve mass

LSFEMs for nonlinear problems

• Consider the nonlinear PDE problem

Lu + G(u) = f in Ω and Ru = g on Γ

where G(u) is a nonlinear term

• Formally, a least-squares principle for linear problems can be easily extendedto handle nonlinear problems

– by posing the problem

minu∈S

JG(u; f, g), where JG(u; f, g) = ‖Lu+G(u)−f‖2HΩ

+‖Ru−g‖2HΓ

– then defining a LSFEM by restricting this problem to a family Sh ⊂ S

– in this way, many nonlinear problems have been solved using LSFEMs

• However, the analysis of LSFEMs for nonlinear problems is not straightforwardand remains one of the open problems in LSFEMs

– such analyses are mostly confined to the Navier-Stokes equations [Bochev,Bochev+1, Bochev+2]

– one of the obstacles faced in the analysis of LSFEMs for a broader class ofnonlinear problems is that after the application of a least-squares principle,the (differentiation) order of the nonlinear term may change

REFERENCES

[Bochev]

Bochev, P., Analysis of least-squares finite element methods for the Navier-Stokesequations, SIAM J. Num. Anal. 34 (1997), 1817–1844

[Bochev+3]

Bochev, P., Cai, Z., Manteuffel, T., McCormick, S., Analysis of velocity-flux leastsquares methods for the Navier-Stokes equations, Part-I, SIAM. J. Num. Anal.

35 (1998) 990–1009

[Bochev & Choi]

Bochev, P., Choi, J., Improved least-squares error estimates for scalar hyperbolicproblems, Comput. Meth. Appl. Math. 1 (2001), 115–124

[Bochev+2]

Bochev, P., Manteuffel, T., McCormick, S., Analysis of velocity-flux least squaresmethods for the Navier-Stokes equations, Part-II, SIAM. J. Num. Anal. 36

(1999) 1125–1144

[Chang & Nelson]

Chang, C.-L., Nelson, J., Least squares finite element method for the Stokesproblem with zero residual of mass conservation, SIAM J. Numer. Anal. 34

(1997), 480–489

[Deang & G.]

Deang, J., Gunzburger, M., Issues related to least-squares finite element methodsfor the Stokes equations, SIAM J. Sci. Comp. 20 (1998), 878–906

[Guermond]

Guermond, J.-L., A finite element technique for solving first-order PDEs in Lp,

SIAM J. Num. Anal. 42 (2004), 714–737

[Jiang]

Jiang, B.-N., Non-oscillatory and non-diffusive solution of convection problemsby the iteratively reweighted least-squares finite element method, J. Comp.

Phys. 105 (1993), 108–121