Robust Algebraic Multilevel Methods And Algorithms

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ISBN: 9783110193657 | 3110193655
Cover: Hardcover
Copyright: 9/15/2009

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This book deals with algorithms for the solution of linear systems of algebraic equations with large-scale sparse matrices, with a focus on problems that are obtained after discretization of partial differential equations using finite element methods. Provides a systematic presentation of the recent advances in robust algebraic multilevel methods. Can be used for advanced courses on the topic.

Preface	p. v
Introduction	p. 1
Finite element method (FEM)	p. 1
The elliptic model problem	p. 1
Conforming discretizations	p. 4
Nonconforming discretizations	p. 5
Structure and properties of the stiffness matrix	p. 8
Ill-conditioned problems	p. 10
Anisotropic problems	p. 10
Problems with highly varying coefficients	p. 10
Elastic deformation of almost incompressible materials	p. 11
High-Reynolds-number flow	p. 12
Preliminaries on iterative methods	p. 13
Stationary methods	p. 13
Polynomial acceleration	p. 15
Conjugate gradients (CG)	p. 18
From steepest descent to conjugate gradients	p. 18
Convergence analysis of the CG method	p. 20
Preconditioned conjugate gradient (PCG) method	p. 24
Generalized conjugate gradient (GCG) method	p. 26
Algebraic multilevel iteration methods	p. 29
Block-factorization: Schur complement	p. 29
Local estimates of the CBS constant	p. 31
Two-level preconditioning methods	p. 34
Algebraic two-level methods	p. 34
Two-level preconditioners for FEM systems	p. 37
Linear AMLI methods	p. 38
Nonlinear AMLI methods	p. 43
Optimality conditions	p. 48
Robustness of the AMLI methods	p. 51
Local analysis of the model problem	p. 51
Robust preconditioning strategy	p. 53
Hierarchical error estimators	p. 54
Robust AMLI algorithms: Conforming linear finite elements	p. 57
Some basic relations	p. 57
Uniform estimates of the constant in the strengthened CBS inequality	p. 62
Additive preconditioning of the pivot block	p. 66
Multiplicative preconditioning of the pivot block	p. 69
Locally improved estimates of the AMLI parameters	p. 72
Optimal complexity solution algorithms for systems with C₁₁	p. 73
A model problem	p. 73
Additive algorithm	p. 74
Multiplicative algorithm	p. 74
Robust AMLI algorithms: Nonconforming linear finite elements	p. 77
Crouzeix-Raviart finite elements	p. 77
Two-level splittings: "First Reduce" and "Differences and Aggregates"	p. 79
Uniform estimates of the CBS constant in case of non-nested spaces	p. 81
Preconditioning of the pivot block	p. 95
Numerical results	p. 98
Concluding remarks	p. 100
Schur complement based multilevel preconditioners	p. 102
Hierarchical versus standard nodal-basis	p. 102
A general two-level preconditioner	p. 102
Incomplete factorization based on exact local factorization	p. 107
Local Schur complements	p. 112
Numerical results	p. 114
Algebraic multigrid (AMG)	p. 117
Two-grid and multigrid algorithms	p. 118
Exact two-level method	p. 118
From two-grid to multigrid	p. 119
Main components of algebraic multigrid	p. 120
Coarse-grid correction	p. 120
Smoothing	p. 121
Interpolation	p. 123
A simple convergence result	p. 125
Error propagation of AMG and AMLI methods	p. 127
Classical AMG	p. 131
Strong connections	p. 131
Coarse-grid selection	p. 131
Interpolation	p. 132
Smoothed aggregation and adaptive AMG methods	p. 132
Utilizing AMG components in AMLI	p. 134
Preconditioning of Rannacher-Turek nonconforming FE systems	p. 136
Rannacher-Turek nonconforming FE systems	p. 136
The nonconforming FE problem	p. 136
Rotated bilinear elements	p. 137
Rotated trilinear elements	p. 138
Hierarchical two-level splittings: 2D case	p. 140
First reduce two-level splitting	p. 140
Two-level splitting by differences and aggregates	p. 143
Uniform estimates of the CBS constant for the 2D splittings	p. 144
Hierarchical two-level splittings: 3D case	p. 147
First reduce two-level splitting	p. 147
Two-level splitting by differences and aggregates	p. 150
Uniform estimates of the CBS constant for the 3D splittings	p. 152
Multilevel preconditioning	p. 155
Numerical tests	p. 158
Additive and multiplicative AMLI preconditioners in 2D	p. 158
Problems with jumping coefficients in 3D	p. 159
AMLI algorithms for discontinuous Galerkin FE problems	p. 164
Introduction to discontinuous Galerkin FEM	p. 164
Element-based approach: bilinear DG systems	p. 166
Face-based approach: Rotated bilinear DG systems	p. 177
Two-level method and AMLI preconditioning of graph-Laplacians	p. 186
AMLI methods for coupled problems	p. 196
AMLI preconditioning of linear elasticity problems	p. 196
Lamé system of elasticity	p. 196
On the robustness of AMLI for conforming FE elasticity systems	p. 198
Locking-free AMLI methods for Crouzeix-Raviart FE discretization of the pure displacement elasticity problem	p. 205
Optimal order AMLI preconditioning of the Navier-Stokes problem	p. 211
Crouzeix-Raviart FE discretization of the velocity field	p. 211
AMLI preconditioning of the mixed FE system: weighted graph-Laplacian	p. 213
Practical issues	p. 219
Linear AMLI algorithm	p. 219
Nonlinear AMLI algorithm	p. 223
Case study: Integrating of new AMLI solvers	p. 225
Crouzeix-Raviart FE discretization of 3D pure displacement elasticity problems	p. 226
Composite FR algorithm	p. 228
Numerical tests: Towards ¿FEM analysis of bone structures	p. 232
Bibliography	p. 237
Index	p. 245
Table of Contents provided by Ingram. All Rights Reserved.

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Robust Algebraic Multilevel Methods And Algorithms

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Robust Algebraic Multilevel Methods And Algorithms

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Author Biography

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