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15th Internet Seminar 2011/12

Operator Semigroups for Numerical Analysis

The 15th Internet Seminar on Evolution Equations is devoted to operator semigroup methods for numerical analysis. Based on the Lax Equivalence Theorem we give an oper- ator theoretic and functional analytic approach to the numerical treatment of evolution equations.

The lectures are at a beginning graduate level and only assume basic familiarity with functional analysis, ordinary and partial differential equations, and numerical analysis.

Organised by the European consortium “International School on Evolution Equations”, the annual Internet Seminars introduce master-, Ph.D. students and postdocs to varying subjects related to evolution equations. The course consists of three phases.

• In Phase 1 (October-February), a weekly lecture will be provided via the ISEM website. Our aim is to give a thorough introduction to the field, at a speed suitable for master’s or Ph.D. students. The weekly lecture will be accompanied by exercises, and the participants are supposed to solve these problems.

• In Phase 2 (March-May), the participants will form small international groups to work on diverse projects which complement the theory of Phase 1 and provide some applications of it.

• Finally, Phase 3 (3-9 June 2012) consists of the final one-week workshop at the Heinrich–Fabri Institut in Blaubeuren (Germany). There the teams will present their projects and additional lectures will be delivered by leading experts.

ISEM team 2011/12:

Virtual lecturers: András Bátkai (Budapest) Bálint Farkas (Budapest) Petra Csomós (Innsbruck) Alexander Ostermann (Innsbruck)

Website: https://isem-mathematik.uibk.ac.at

Further Information: [email protected]

Description of the course

The course concentrates on the numerical solution of initial value problems of the type

u′(t) = Au(t) + f(t), t ≥ 0, u(0) = u0 ∈ D(A),

where A is a linear operator with dense domain of definition D(A) in a Banach space X, and u0 is the initial value. A model example is the Laplace operator A = ∆ with appropriate domain in the Hilbert space L2(Ω). In this case the above partial differential equation describes heat conduction inside Ω. One way of finding a solution to this initial value problem is to imitate the way in which one solves linear ordinary differential equations with constant coefficients: First define the exponential etA in suitable way. Then the solution of the homogeneous problem is given by this fundamental operator applied to the initial value u0, i.e., u(t) = e

tAu0. This is where operator semigroup theory enters the game: the fundamental operators T (t) := etA form a so-called strongly continuous semigroup of bounded linear operators on the Banach space X. That is to say the functional equation T (t+s) = T (t)T (s) and T (0) = I holds together with the continuity of the orbits t 7→ T (t)u0. If such a semigroup exists, we say that the initial value problem is well-posed. Once existence and uniqueness of solutions are guaranteed, the following numerical aspects appear.

• In most cases the operator A is complicated and numerically impossible to work with, so one approximates it via a sequence of (simple) operators Am hoping that the corresponding solutions e

tAm (expected to be easily computable) converge to the solution of the original problem etA in some sense. This procedure is called space discretisation. This discretisation may indeed come from a spatial mesh (e.g., for a finite difference method) or from some not so space-related discretisations, e.g., from Fourier-Galerkin methods.

• Equally hard is the computation of the exponential of an operator A. One idea is to approximate the exponential function z 7→ ez by functions r that are easier to handle. A typical example, known also from basic calculus courses, is the backward Euler scheme r(z) = (1 − z)−1. In this case the approximation means r(0) = r′(0) = e0, i.e., the first two Taylor coefficients of r and of the exponential function coincide. This leads to the following idea. If r(tA) is approximately the same as etA for small values of t (up to an error of magnitude t2), we may take the nth power of it. To compensate for the growing error, we take decreasing time steps as n grows and obtain[

r( tnA) ]n ≈ [e tnA]n = etA

by the semigroup property. This procedure is called temporal discretisation.

• Due to numerical reasons, one is usually forced to combine the above two methods and add further spice to the stew: operator splitting. This is usually done when the operator A has a complicated structure, but decomposes into a finite number of parts that are easier to handle.

In semigroup theory the above methods culminate in the famous Lax Equivalence Theorem and Chernoff’s Theorem, describing precisely the situation when these methods work. In this course we shall develop the basic tools from operator semigroup theory needed for such an abstract treatment of discretisation procedures.

Topics to be covered include:

1 initial value problems and operator semigroups,

1 spatial discretisations, Trotter–Kato theorems, finite element and finite difference approximations,

1 fractional powers, interpolation spaces, analytic semigroups,

1 the Lax Equivalence Theorem and Chernoff’s Theorem, error estimates, order of convergence, stability issues,

1 temporal discretisations, rational approximations, Runge–Kutta methods, operator splitting procedures,

1 applications to various differential equations, like inhomogeneous problems, non-autonomous equations, semilinear equations, Schrödinger equations, delay differential equations, Volterra equations,

1 exponential integrators.

Some of these topics will be elaborated on in Phase 2, where the students will have the possibility to work on projects which are related to active research.

Lecture 1

What is the Topic of this Course?

The ultimate aim of these notes is quickly formulated: We would like to develop those functional analytic tools that allows us to adopt methods for ordinary differential equations (ODEs) to solve some classes of time-dependent partial differential equations (PDEs) numerically.

Let us illustrate this idea by recalling first the most trivial one of all ODEs. For a matrix A ∈ ℝd×d

consider the initial value problem

{

u̇(t) = Au(t),

u(0) = u0.

We know that the solution to such an ordinary differential equation is given by

u(t) = etAu0,

where etA is the exponential function of the matrix tA defined by the power series

etA = ∞ ∑

n=0

tnAn

n! ,

which converges absolutely and uniformly on every compact interval of ℝ. Here the numerical challenge is, especially for large matrices, to calculate this exponential function in an effective and accurate way.

The exponential function of a matrix plays an important role not only because it solves the linear problem above, but it also occurs in more complicated problems where a nonlinearity is present, like in the equation

v̇(t) = Av(t) + F (t, v(t)).

To solve such an equation by iterative methods the variation of constants formula plays an essential role, stating that the solution v(t) of this nonlinear equation satisfies

v(t) = etAv(0) +

t ∫

0

e(t−s)AF (s, v(s)) ds.

Here again the exponential function of a matrix appears. Of course, here further numerical issues arise, such as the calculation of integrals.

There is a multitude of theoretical methods for the calculation of such exponentials, each of them leading to some possible numerical treatment of the problem. We mention those that will be important for us in this course:

1. by means of the Jordan normal form,

1

2 Lecture 1: What is the Topic of this Course?

2. by means of the Cauchy’s integral formula, more precisely, by using the identity

1

2�i

∮

e�

�− z d� = ez,

3. by using other formulae for the exponential function, say

ex = lim n→∞

(

1 + x

n

)n

= lim n→∞

(

1− x

n

)

−n

.

Let us start by looking at the first of the suggestions on the above list. Theory tells us that we “only” have to bring A to Jordan normal form, and then the exponential function can be simply read off. The situation is even better if we can find a basis of orthogonal eigenvectors. Then we can bring the matrix A to diagonal form by a similarity transformation S−1AS = D = diag(�1, . . . , �d), and hence the exponential becomes

etA = SetDS−1 = S diag(et�1 , . . . , et�d)S−1.

Of course, other numerical difficulties are hidden in calculating the Jordan normal form or the similarity transformation S. Still this very idea proves itself to be useful for partial differential equations. Let us illustrate this idea on the next example.

1.1 The heat equation

Consider the one-dimensional heat equation, say, on the interval (0, �)

∂tw(t, x) = ∂xxw(t, x), t > 0

w(0, x) = w0(x),

with homogeneous Dirichlet boundary conditions

w(t, 0) = w(t, �) = 0.

We can rewrite this equation (without the initial condition) as a linear ordinary differential equation

u̇(t) = Au(t), t > 0 (1.1)

in the infinite dimensional Hilbert space L2(0, �). To do this define the operator

(Ag)(x) := g′′(x) = d2

dx2 g(x)