Discretization Methods and Iterative Solvers Based on Domain Decomposition : Lecture Notes in Computational Science and Engineering - Barbara I. Wohlmuth

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Discretization Methods and Iterative Solvers Based on Domain Decomposition

By: Barbara I. Wohlmuth

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Paperback | 27 February 2001

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Domain decomposition methods provide powerful and flexible tools for the numerical approximation of partial differential equations arising in the modeling of many interesting applications in science and engineering. This book deals with discretization techniques on non-matching triangulations and iterative solvers with particular emphasis on mortar finite elements, Schwarz methods and multigrid techniques. New results on non-standard situations as mortar methods based on dual basis functions and vector field discretizations are analyzed and illustrated by numerical results. The role of trace theorems, harmonic extensions, dual norms and weak interface conditions is emphasized. Although the original idea was used successfully more than a hundred years ago, these methods are relatively new for the numerical approximation. The possibilites of high performance computations and the interest in large- scale problems have led to an increased research activity.

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Preface	p. v
Discretization Techniques Based on Domain Decomposition	p. 1
Introduction to Mortar Finite Element Methods	p. 3
Mortar Methods with Alternative Lagrange Multiplier Spaces	p. 11
An Approximation Property	p. 15
The Consistency Error	p. 19
Discrete Inf-sup Conditions	p. 24
Examples of Lagrange Multiplier Spaces	p. 27
The First Order Case in 2D	p. 29
The First Order Case in 3D	p. 33
The Second Order Case in 2D	p. 36
Discretization Techniques Based on the Product Space	p. 37
A Dirichlet{Neumann Formulation	p. 40
Variational Formulations	p. 43
Algebraic Formulations	p. 47
Examples for Special Mortar Finite Element Discretizations	p. 50
The Coupling of Primal and Dual Finite Elements	p. 51
An Equivalent Nonconforming Formulation	p. 54
Crouzeix{Raviart Finite Elements	p. 58
Numerical Results	p. 61
In uence of the Lagrange Multiplier Spaces	p. 64
A Non-optimal Mortar Method	p. 71
In uence of the Choice of the Mortar Side	p. 74
In uence of the Jump of the CoeÆcients	p. 83
Iterative Solvers Based on Domain Decomposition	p. 85
Abstract Schwarz Theory	p. 87
Additive Schwarz Methods	p. 88
Multiplicative Schwarz Methods	p. 95
Multigrid Methods	p. 97
Vector Field Discretizations	p. 99
Raviart{Thomas Finite Elements	p. 101
An Iterative Substructuring Method	p. 103
An Interpolation Operator onto VH	p. 105
An Extension Operator onto VF	p. 107
Quasi-optimal Bounds	p. 113
A Hierarchical Basis Method	p. 114
Horizontal Decomposition	p. 115
Vertical Decomposition	p. 118
Numerical Results	p. 121
The 2D Case	p. 122
The 3D Case	p. 123
A Multigrid Method for the Mortar Product Space Formulation	p. 125
Bilinear Forms	p. 126
An Approximation Property	p. 129
Smoothing and Stability Properties	p. 131
Implementation of the Smoothing Step	p. 136
Numerical Results in 2D and 3D	p. 137
Extensions to Linear Elasticity	p. 142
Uniform Ellipticity	p. 145
Numerical Results	p. 149
A Weaker Interface Condition	p. 151
A Dirichlet{Neumann Type Method	p. 155
The Algorithm	p. 155
Numerical Results	p. 158
A Multigrid Method for the Mortar Saddle Point Formulation	p. 162
An Approximation Property	p. 164
Smoothing and Stability Properties	p. 167
A Block Diagonal Smoother	p. 168
An Inde nite Smoother	p. 171
Numerical Results	p. 174
Bibliography	p. 177
List of Figures	p. 187
List of Tables	p. 189
Notations	p. 191
Index	p. 195
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Discretization Methods and Iterative Solvers Based on Domain Decomposition

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