Adaptive Finite Element Methods for Differential Equations

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Springer Science & Business Media, Jan 23, 2003 - Mathematics - 207 pages
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These Lecture Notes have been compiled from the material presented by the second author in a lecture series ('Nachdiplomvorlesung') at the Department of Mathematics of the ETH Zurich during the summer term 2002. Concepts of 'self adaptivity' in the numerical solution of differential equations are discussed with emphasis on Galerkin finite element methods. The key issues are a posteriori er ror estimation and automatic mesh adaptation. Besides the traditional approach of energy-norm error control, a new duality-based technique, the Dual Weighted Residual method (or shortly D WR method) for goal-oriented error estimation is discussed in detail. This method aims at economical computation of arbitrary quantities of physical interest by properly adapting the computational mesh. This is typically required in the design cycles of technical applications. For example, the drag coefficient of a body immersed in a viscous flow is computed, then it is minimized by varying certain control parameters, and finally the stability of the resulting flow is investigated by solving an eigenvalue problem. 'Goal-oriented' adaptivity is designed to achieve these tasks with minimal cost. The basics of the DWR method and various of its applications are described in the following survey articles: R. Rannacher [114], Error control in finite element computations. In: Proc. of Summer School Error Control and Adaptivity in Scientific Computing (H. Bulgak and C. Zenger, eds), pp. 247-278. Kluwer Academic Publishers, 1998. M. Braack and R. Rannacher [42], Adaptive finite element methods for low Mach-number flows with chemical reactions.
 

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Contents

Introduction
1
computation of drag coefficient
2
12 The need for goaloriented mesh adaptation
4
13 Further examples of goaloriented simulation
9
14 General concepts of error estimation
11
An ODE Model Case
15
FD versus FE method
19
23 Exercises
23
71 A posteriori error analysis
82
72 Error control for functionals of eigenfunctions
91
73 The stability eigenvalue problem
95
74 Exercises
99
Optimization Problems
101
81 A posteriori error analysis via Lagrange formalism
103
82 Application to a boundary control problem
105
83 Application to parameter estimation
110

A PDE Model Case
25
31 Finite element approximation
26
32 Global a posteriori error estimates
29
33 A posteriori error estimates for output functionals
30
34 Higherorder finite elements
37
35 Exercises
39
Practical Aspects
41
41 Evaluation of the error identity and indicators
42
42 Mesh adaptation
46
43 Use of error estimators for postprocessing
52
44 Towards anisotropic mesh adaptation
55
45 Exercises
60
The Limits of Theoretical Analysis
61
51 Convergence of residuals
64
52 Approximation of weights
65
53 Exercises
69
An Abstract Approach for Nonlinear Problems
71
61 Galerkin approximation of nonlinear equations
72
62 A nested solution approach
78
63 Exercises
79
Eigenvalue Problems
81
84 Exercises
111
TimeDependent Problems
113
the heat equation
115
the wave equation
123
94 Exercises
128
Applications in Structural Mechanics
129
102 A model problem in elastoplasticity theory
134
103 Exercises
142
Applications in Fluid Mechanics
143
111 Computation of drag and lift in a viscous flow
144
112 Minimization of drag by boundary control
152
113 Stability analysis for stationary flow
156
114 Exercises
160
Miscellaneous and Open Problems
161
122 Current developments
162
123 Open problems
164
Solutions of exercises
167
Bibliography
191
Index
203
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About the author (2003)

Rannacher is of the Universitat Heidelberg.

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