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Table of Contents
Intro
Contents
1 Introduction
2 Mixed Problems
2.1 Some Reminders About Mixed Problems
2.1.1 The Saddle Point Formulation
2.1.2 Existence of a Solution
2.1.3 Dual Problem
2.1.4 A More General Case: A Regular Perturbation
2.1.5 The Case
2.2 The Discrete Problem
2.2.1 Error Estimates
2.2.2 The Matricial Form of the Discrete Problem
2.2.3 The Discrete Dual Problem: The Schur Complement
2.3 Augmented Lagrangian
2.3.1 Augmented or Regularised Lagrangians
2.3.2 Discrete Augmented Lagrangian in Matrix Form
2.3.3 Augmented Lagrangian and the Condition Number of the Dual Problem
2.3.4 Augmented Lagrangian: An Iterated Penalty
3 Iterative Solvers for Mixed Problems
3.1 Classical Iterative Methods
3.1.1 Some General Points
Linear Algebra and Optimisation
Norms
Krylov Subspace
Preconditioning
3.1.2 The Preconditioned Conjugate Gradient Method
3.1.3 Constrained Problems: Projected Gradient and Variants
Equality Constraints: The Projected Gradient Method
Inequality Constraints
Positivity Constraints
Convex Constraints
3.1.4 Hierarchical Basis and Multigrid Preconditioning
3.1.5 Conjugate Residuals, Minres, Gmres and the Generalised Conjugate Residual Algorithm
The Generalised Conjugate Residual Method
The Left Preconditioning
The Right Preconditioning
The Gram-Schmidt Algorithm
GCR for Mixed Problems
3.2 Preconditioners for the Mixed Problem
3.2.1 Factorisation of the System
Solving Using the Factorisation
3.2.2 Approximate Solvers for the Schur Complement and the Uzawa Algorithm
The Uzawa Algorithm
3.2.3 The General Preconditioned Algorithm
3.2.4 Augmented Lagrangian as a Perturbed Problem
4 Numerical Results: Cases Where Q= Q
4.1 Mixed Laplacian Problem
4.1.1 Formulation of the Problem
4.1.2 Discrete Problem and Classic Numerical Methods
The Augmented Lagrangian Formulation
4.1.3 A Numerical Example
4.2 Application to Incompressible Elasticity
4.2.1 Nearly Incompressible Linear Elasticity
4.2.2 Neo-Hookean and Mooney-Rivlin Materials
Mixed Formulation for Mooney-Rivlin Materials
4.2.3 Numerical Results for the Linear Elasticity Problem
4.2.4 The Mixed-GMP-GCR Method
Approximate Solver in u
4.2.5 The Test Case
Number of Iterations and Mesh Size
Comparison of the Preconditioners of Sect.3.2
Effect of the Solver in u
4.2.6 Large Deformation Problems
Neo-Hookean Material
Mooney-Rivlin Material
4.3 Navier-Stokes Equations
4.3.1 A Direct Iteration: Regularising the Problem
4.3.2 A Toy Problem
5 Contact Problems: A Case Where Q=Q
5.1 Imposing Dirichlet's Condition Through a Multiplier
5.1.1 and Its Dual
5.1.2 A Steklov-Poincaré operator
Using This as a Solver
5.1.3 Discrete Problems
The Matrix Form and the Discrete Schur Complement
Contents
1 Introduction
2 Mixed Problems
2.1 Some Reminders About Mixed Problems
2.1.1 The Saddle Point Formulation
2.1.2 Existence of a Solution
2.1.3 Dual Problem
2.1.4 A More General Case: A Regular Perturbation
2.1.5 The Case
2.2 The Discrete Problem
2.2.1 Error Estimates
2.2.2 The Matricial Form of the Discrete Problem
2.2.3 The Discrete Dual Problem: The Schur Complement
2.3 Augmented Lagrangian
2.3.1 Augmented or Regularised Lagrangians
2.3.2 Discrete Augmented Lagrangian in Matrix Form
2.3.3 Augmented Lagrangian and the Condition Number of the Dual Problem
2.3.4 Augmented Lagrangian: An Iterated Penalty
3 Iterative Solvers for Mixed Problems
3.1 Classical Iterative Methods
3.1.1 Some General Points
Linear Algebra and Optimisation
Norms
Krylov Subspace
Preconditioning
3.1.2 The Preconditioned Conjugate Gradient Method
3.1.3 Constrained Problems: Projected Gradient and Variants
Equality Constraints: The Projected Gradient Method
Inequality Constraints
Positivity Constraints
Convex Constraints
3.1.4 Hierarchical Basis and Multigrid Preconditioning
3.1.5 Conjugate Residuals, Minres, Gmres and the Generalised Conjugate Residual Algorithm
The Generalised Conjugate Residual Method
The Left Preconditioning
The Right Preconditioning
The Gram-Schmidt Algorithm
GCR for Mixed Problems
3.2 Preconditioners for the Mixed Problem
3.2.1 Factorisation of the System
Solving Using the Factorisation
3.2.2 Approximate Solvers for the Schur Complement and the Uzawa Algorithm
The Uzawa Algorithm
3.2.3 The General Preconditioned Algorithm
3.2.4 Augmented Lagrangian as a Perturbed Problem
4 Numerical Results: Cases Where Q= Q
4.1 Mixed Laplacian Problem
4.1.1 Formulation of the Problem
4.1.2 Discrete Problem and Classic Numerical Methods
The Augmented Lagrangian Formulation
4.1.3 A Numerical Example
4.2 Application to Incompressible Elasticity
4.2.1 Nearly Incompressible Linear Elasticity
4.2.2 Neo-Hookean and Mooney-Rivlin Materials
Mixed Formulation for Mooney-Rivlin Materials
4.2.3 Numerical Results for the Linear Elasticity Problem
4.2.4 The Mixed-GMP-GCR Method
Approximate Solver in u
4.2.5 The Test Case
Number of Iterations and Mesh Size
Comparison of the Preconditioners of Sect.3.2
Effect of the Solver in u
4.2.6 Large Deformation Problems
Neo-Hookean Material
Mooney-Rivlin Material
4.3 Navier-Stokes Equations
4.3.1 A Direct Iteration: Regularising the Problem
4.3.2 A Toy Problem
5 Contact Problems: A Case Where Q=Q
5.1 Imposing Dirichlet's Condition Through a Multiplier
5.1.1 and Its Dual
5.1.2 A Steklov-Poincaré operator
Using This as a Solver
5.1.3 Discrete Problems
The Matrix Form and the Discrete Schur Complement