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Table of contents
Resume
Abstract v
Acknowledgements
List of Figures
List of Tables
Introduction
i Finite element method
ii Linear solvers
ii.1 Direct solvers
ii.2 Iterative solvers
ii.3 Adaptive linear solvers
ii.4 Robustness with respect to the polynomial degree p
iii Two central building blocks for the results of the thesis
iv Model problem and its discretization
v A posteriori point of view and goals of the thesis
v.1 Multilevel setting
v.2 Multilevel procedure for constructing an a posteriori estimator of the algebraic error and a linear solver
v.3 Main results: p-robust eciency of the a posteriori estimator and p-robust solver contraction
v.4 Alternative approaches
v.5 Extension to the mixed nite element method
vi Adaptivity in a-posteriori-steered solvers
vi.1 Adaptive number of post-smoothing steps
vi.2 Adaptive local smoothing
vii Contents and contributions of the thesis
vii.1 Chapter 1
vii.2 Chapter 2
vii.3 Chapter 3
vii.4 Chapter 4
vii.5 Implementation notes
viii Perspectives
1 A multilevel algebraic error estimator and the corresponding iterative solver with p-robust behavior
1 Introduction
2 Setting
2.1 Model problem
2.2 Finite element discretization
2.3 Algebraic system, approximate solution, and algebraic residual
2.4 A hierarchy of meshes
2.5 A hierarchy of spaces
2.6 Two types of patches
3 Multilevel lifting of the algebraic residual
3.1 Exact algebraic residual lifting
3.2 Coarse solve
3.3 Multilevel algebraic residual lifting
4 An a posteriori estimator on the algebraic error and a multilevel solver
4.1 A posteriori estimate on the algebraic error
4.2 Multilevel solver
5 Main results
6 Numerical experiments
6.1 Performance of the damped additive Schwarz (dAS) construction of the solver
6.2 Performance of the weighted restrictive additive Schwarz (wRAS) construction of the solver
6.3 Comparison with other multilevel solvers
7 Proofs of the main results
7.1 Upper bound on kri J;algk
7.2 Lower bound on (f; i J;alg) (rui J ;ri J;alg)
7.3 Polynomial-degree-robust multilevel stable decomposition
7.4 Upper bound on kr~i J;algk
7.5 Proof of Theorem 5.1
7.6 Proof of Corollary 5.6
8 Conclusions and outlook
2 A-posteriori-steered p-robust multigrid with optimal step-sizes and adaptive number of smoothing steps
1 Introduction
2 Setting
2.1 Model problem, nite element discretization, and algebraic system
2.2 A hierarchy of meshes and spaces
3 Motivation: level-wise orthogonal decomposition of the error
4 Multilevel solver
5 A posteriori estimator on the algebraic error
6 Main results
6.1 Setting, mesh, and regularity assumptions
6.2 Main results
6.3 Additional results
7 Adaptive number of smoothing steps
8 Complexity of the solver
9 Numerical experiments
9.1 Performance of the multilevel solver of Denition 4.1
9.2 Adaptive number of smoothing steps using Denition 7.1
9.3 Examples in three space dimensions
9.4 Comparison with solvers from literature
10 Proof of Theorem 6.6
10.1 Properties of the estimator i alg
10.2 Properties of the exact residual lifting ~i J;alg
10.3 Proof of Theorem 6.6 under the minimal H1 0 ( )-regularity assumption
10.4 Proof of Theorem 6.6 under the H2( )-regularity assumption
11 Conclusions and future work
3 Contractive local adaptive smoothing based on Dor er’s marking in aposteriori- steered p-robust multigrid solvers
1 Introduction
2 Setting
2.1 Model problem and its nite element discretization
2.2 A hierarchy of meshes and spaces
3 Adaptive multilevel solver
3.1 Algorithmic description of the solver
3.2 Mathematical description of the solver
4 A posteriori estimator on the algebraic error
5 Main results
5.1 Mesh assumptions
5.2 Main result
5.3 Additional results
6 Numerical experiments
6.1 Can we predict the distribution of the algebraic error?
6.2 Does the adaptivity pay o?
6.3 Dependence on the marking parameter
7 Proofs of the main results
7.1 Proof of contraction: full-smoothing substep
7.2 Proof of contraction: adaptive-smoothing substep
8 Conclusions
4 p-robust multilevel and domain decomposition methods with optimal stepsizes for mixed nite element discretizations of elliptic problems
1 Introduction
2 Model problem and its mixed nite element discretization
2.1 Discrete mixed nite element problem
3 Multilevel setting
3.1 A hierarchy of meshes
3.2 A hierarchy of spaces
4 An a-posteriori-steered multigrid solver
4.1 Multigrid solver
4.2 A posteriori estimator on the algebraic error
5 An a-posteriori-steered domain decomposition solver
5.1 Two-level hierarchy
5.2 Two-level iterative solver: overlapping additive Schwarz .
5.3 A posteriori estimator on the algebraic error
6 Main results
7 Proofs of the main results
7.1 Multilevel setting results
7.2 Two-level domain decomposition setting results
8 Conclusions
