Galerkin Finite Element Methods for Parabolic Problems - Vidar Thomee.pdf - MES(1) - Sygnity

S pringer Series in

C omputational

M athematics

Editorial Board

R. Bank

R.L. Graham

J. Stoer

R. Varga

H. Yserentant

Vidar Thomée

Galerkin Finite Element

Methods for Parabolic

Problems

Second Edition

ABC

Vidar Thomée

Department of Mathematics

Chalmers University of Technology

S-41296 Göteborg

Sweden

email: thomee@math.chalmers.se

Library of Congress Control Number: 2006925896

Mathematics Subject Classiﬁcation (2000): 65M60, 65M12, 65M15

ISSN 0179-3632

ISBN-10 3-540-33121-2 Springer Berlin Heidelberg New York

ISBN-13 978-3-540-33121-6 Springer Berlin Heidelberg New York

ISBN-10 3-540-63236-0 1st Edition Springer Berlin Heidelberg New York

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Preface

My purpose in this monograph is to present an essentially self-contained

account of the mathematical theory of Galerkin ﬁnite element methods as

applied to parabolic partial diﬀerential equations. The emphases and selection

of topics reﬂects my own involvement in the ﬁeld over the past 25 years,

and my ambition has been to stress ideas and methods of analysis rather

than to describe the most general and farreaching results possible. Since the

formulation and analysis of Galerkin ﬁnite element methods for parabolic

problems are generally based on ideas and results from the corresponding

theory for stationary elliptic problems, such material is often included in the

presentation.

The basis of this work is my earlier text entitled Galerkin Finite Element

Methods for Parabolic Problems, Springer Lecture Notes in Mathematics,

No. 1054, from 1984. This has been out of print for several years, and I

have felt a need and been encouraged by colleagues and friends to publish an

updated version. In doing so I have included most of the contents of the 14

chapters of the earlier work in an updated and revised form, and added four

new chapters, on semigroup methods, on multistep schemes, on incomplete

iterative solution of the linear algebraic systems at the time levels, and on

semilinear equations. The old chapters on fully discrete methods have been

reworked by ﬁrst treating the time discretization of an abstract diﬀerential

equation in a Hilbert space setting, and the chapter on the discontinuous

Galerkin method has been completely rewritten.

The following is an outline of the contents of the book:

In the introductory Chapter 1 we begin with a review of standard material

on the ﬁnite element method for Dirichlets problem for Poissons equation

in a bounded domain, and consider then the simplest Galerkin ﬁnite element

methods for the corresponding initial-boundary value problem for the linear

heat equation. The discrete methods are based on associated weak, or vari-

ational, formulations of the problems and employ ﬁrst piecewise linear and

then more general approximating functions which vanish on the boundary

of the domain. For these model problems we demonstrate the basic error

estimates in energy and mean square norms, in the parabolic case ﬁrst for

the semidiscrete problem resulting from discretization in the spatial vari-

ables only, and then also for the most commonly used fully discrete schemes

Preface

obtained by discretization in both space and time, such as the backward Euler

and Crank-Nicolson methods.

In the following ﬁve chapters we study several extensions and generaliza-

tions of the results obtained in the introduction in the case of the spatially

semidiscrete approximation, and show error estimates in a variety of norms.

First, in Chapter 2, we formulate the semidiscrete problem in terms of a more

general approximate solution operator for the elliptic problem in a manner

which does not require the approximating functions to satisfy the homoge-

neous boundary conditions. As an example of such a method we discuss a

method of Nitsche based on a nonstandard weak formulation. In Chapter 3

more precise results are shown in the case of the homogeneous heat equation.

These results are expressed in terms of certain function spaces H s (Ω)which

are characterized by both smoothness and boundary behavior of its elements,

and which will be used repeatedly in the rest of the book. We also demon-

strate that the smoothing property for positive time of the solution operator

of the initial value problem has an analogue in the semidiscrete situation, and

use this to show that the ﬁnite element solution converges to full order even

when the initial data are nonsmooth. The results of Chapters 2 and 3 are

extended to more general linear parabolic equations in Chapter 4. Chapter

5 is devoted to the derivation of stability and error bounds with respect to

the maximum-norm for our plane model problem, and in Chapter 6 negative

norm error estimates of higher order are derived, together with related results

concerning superconvergence.

In the next six chapters we consider fully discrete methods obtained by

discretization in time of the spatially semidiscrete problem. First, in Chapter

7, we study the homogeneous heat equation and give analogues of our pre-

vious results both for smooth and for nonsmooth data. The methods used

for time discretization are of one-step type and rely on rational approxima-

tions of the exponential, allowing the standard Euler and Crank-Nicolson

procedures as special cases. Our approach here is to ﬁrst discretize a par-

abolic equation in an abstract Hilbert space framework with respect to time,

and then to apply the results obtained to the spatially semidiscrete problem.

The analysis uses eigenfunction expansions related to the elliptic operator

occurring in the parabolic equation, which we assume positive deﬁnite. In

Chapter 8 we generalize the above abstract considerations to a Banach space

setting and allow a more general parabolic equation, which we now analyze

using the Dunford-Taylor spectral representation. The time discretization is

interpreted as a rational approximation of the semigroup generated by the

elliptic operator, i.e., the solution operator of the initial-value problem for

the homogeneous equation. Application to maximum-norm estimates is dis-

cussed. In Chapter 9 we study fully discrete one-step methods for the inho-

mogeneous heat equation in which the forcing term is evaluated at a ﬁxed

ﬁnite number of points per time stepping interval. In Chapter 10 we apply

Galerkins method also for the time discretization and seek discrete solutions

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