Numerical Methods for Mixed Finite Element Problems: Applications to Incompressible Materials and Contact Problems

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
		5.1.4 A Discrete Steklov-Poincaré Operator
		5.1.5 Computational Issues, Approximate Scalar Product
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		5.1.6 The  Formulation
			The Choice of h
		5.1.7 A Toy Model for the Contact Problem
			The Discrete Formulation
			The Active Set Strategy
	5.2 Sliding Contact
		5.2.1 The Discrete Contact Problem
			Contact Status
		5.2.2 The Algorithm for Sliding Contact
			A Newton Method
			The Active Set Strategy
		5.2.3 A Numerical Example of Contact Problem
6 Solving Problems with More Than One Constraint
	6.1 A Model Problem
	6.2 Interlaced Method
	6.3 Preconditioners Based on Factorisation
		6.3.1 The Sequential Method
	6.4 An Alternating Procedure
7 Conclusion
Bibliography
Index