Numerical Solution of Partial Differential Equations by the Finite Element Method

By: CLAES JOHNSON

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Paperback | 27 March 2009

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An accessible introduction to the finite element method for solving numeric problems, this volume offers the keys to an important technique in computational mathematics. Suitable for advanced undergraduate and graduate courses, it outlines clear connections with applications and considers numerous examples from a variety of science- and engineering-related specialties.This text encompasses all varieties of the basic linear partial differential equations, including elliptic, parabolic and hyperbolic problems, as well as stationary and time-dependent problems. Additional topics include finite element methods for integral equations, an introduction to nonlinear problems, and considerations of unique developments of finite element techniques related to parabolic problems, including methods for automatic time step control. The relevant mathematics are expressed in non-technical terms whenever possible, in the interests of keeping the treatment accessible to a majority of students.

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Non-Fiction Mathematics Calculus & Mathematical Analysis Numerical Analysis Differential Calculus & Equations

Preface to the Dover Edition	p. 5
Preface	p. 7
Introduction	p. 9
Background	p. 9
Difference methods - Finite element methods	p. 10
Scope of the book	p. 12
Introduction to FEM for elliptic problems	p. 14
Variational formulation of a one-dimensional model problem	p. 14
FEM for the model problem with piecewise linear functions	p. 18
An error estimate for FEM for the model problem	p. 23
FEM for the Poisson equation	p. 26
The Hilbert spaces L2(¿), H1 (¿) and H10(¿)	p. 33
A geometric interpretation of FEM	p. 38
A Neumann problem. Natural and essential boundary conditions	p. 40
Remarks on programming	p. 43
Remarks on finite element software	p. 48
Abstract formulation of the finite element method for elliptic problems	p. 50
Introduction. The continuous problem	p. 50
Discretization. An error estimate	p. 52
The energy norm	p. 55
Some examples	p. 55
Some finite element spaces	p. 67
Introduction. Regularity requirements	p. 67
Some examples of finite elements	p. 68
Summary	p. 79
Approximation theory for FEM. Error estimates for elliptic problems	p. 84
Introduction	p. 84
Interpolation with piecewise linear functions in two dimensions	p. 84
Interpolation with polynomials of higher degree	p. 90
Error estimates for FEM for elliptic problems	p. 91
On the regularity of the exact solution	p. 92
Adaptive methods	p. 94
An error estimate in the L2(¿)-nom	p. 97
Some applications to elliptic problems	p. 101
The elasticity problem	p. 101
Stokes problem	p. 106
A plate problem	p. 108
Direct methods for solving linear systems of equations	p. 112
Introduction	p. 112
Gaussian elimination. Cholesky's method	p. 112
Operation counts. Band matrices	p. 114
Fill-in	p. 116
The frontal method	p. 117
Nested dissection	p. 120
Minimization algorithms. Iterative methods	p. 123
Introduction	p. 123
The gradient method	p. 128
The conjugate gradient method	p. 131
Preconditioning	p. 136
Multigrid methods	p. 137
Work estimates for direct and iterative methods	p. 139
The condition number of the stiffness matrix	p. 141
FEM for parabolic problems	p. 146
Introduction	p. 146
A one-dimensional model problem	p. 147
Semi-discretization in space	p. 149
Discretization in space and time	p. 152
Background	p. 152
The backward Euler and Crank-Nicolson methods	p. 153
The discontinuous Galerkin method	p. 157
Error estimates for fully discrete approximations and automatic time and space step control	p. 158
Hyperbolic problems	p. 167
Introduction	p. 167
A convection-diffusion problem	p. 168
General remarks on numerical methods for hyperbolic equations	p. 171
Outline and preliminaries	p. 173
Standard Galerkin	p. 176
Classical artificial diffusion	p. 181
The streamline diffusion method	p. 181
The streamline diffusion method with ¿=0	p. 182
The streamline diffusion method with ¿>0	p. 185
The discontinuous Galerkin method	p. 189
The streamline diffusion method for time-dependent convection-diffusion problems	p. 199
Friedrichs' systems	p. 205
The continuous problem	p. 205
The standard Galerkin method	p. 207
The streamline diffusion method	p. 207
The discontinuous Galerkin method	p. 207
Second order hyperbolic problems	p. 210
Boundary element methods	p. 214
Introduction	p. 214
Some integral equations	p. 216
An integral equation for an exterior Dirichlet problem using a single layer potential	p. 219
An exterior Dirichlet problem with double layer potential	p. 220
An exterior Neumann problem with single layer potential	p. 222
Alternative integral equation formulations	p. 223
Finite element methods	p. 224
FEM for a Fredholm equation of the first kind	p. 224
FEM for a Fredholm equation of the second kind	p. 227
Mixed finite element methods	p. 232
Introduction	p. 232
Some examples	p. 234
Curved elements and numerical integration	p. 239
Curved elements	p. 239
Numerical integration (quadrature)	p. 245
Some non-linear problems	p. 248
Introduction	p. 248
Convex minimization problems	p. 248
The continuous problem	p. 248
Discretizations	p. 254
Numerical methods for convex minimization problems	p. 255
A non-linear parabolic problem	p. 257
The incompressible Euler equations	p. 258
The continuous problem	p. 258
The streamline diffusion method in (¿, ¿)-formulation	p. 259
The discontinuous Galerkin method in (¿, ¿)-formulation	p. 260
The streamline diffusion method in (u, p)-formulation	p. 261
The incompressible Navier-Stokes equations	p. 262
Compressible flow: Burgers' equation	p. 263
References	p. 270
Index	p. 276
Table of Contents provided by Ingram. All Rights Reserved.

Numerical Solution of Partial Differential Equations by the Finite Element Method

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