Infotech Oulu Graduate School

Iterative solution methods for mesh variational inequalities

Lecturer: Professor Alexander Lapin, Kazan State University, Russia

Date & time:
Monday, April 20, 2009, 10-12
Tuesday, April 21, 14-16
Wednesday, April 22, 10-12
Thursday, April 23, 12-14
Friday, April 24, 10-12

Room: M242

Credits: 2.0 (with an exercise)

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The  purpose of the lecture course is to give an introduction to a very important field of computational  mathematics -- solution methods for variational inequalities. Iterative methods for finite element  and finite difference (combined in the name  “mesh”) approximations  of variational inequalities, which arise in mathematical physics and optimal control in systems governed by PDEs, are studied. Mesh variational inequalities are divided into two classes:  inequalities with positive definite matrices and saddle-point problems with constraints, or inequalities with “saddle matrices”. The examples of the first class are the mesh approximations of variational inequalities with constraints to a solution, such as obstacle problem and Signorini problem. The approximations of the variational inequalities with constraints to gradient of solution, of the optimal control problems, and mixed hybrid finite element method for variational inequalities lead to the problems of the second class.

Basic iterative methods for the both classes of  inequalities are considered. Convergence and rate of convergence, details of   implementation for iterative methods and algorithms are discussed.

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Contents

1. Mesh variational inequalities with positive definite matrices.

Finite element and/or finite difference approximations of  the variational inequalities with simple constraints to a solution or with non-differentiable functionals (the obstacle problem,  Signorini problem, model elasticity problem).
Elements of convex analysis and equivalent  formulations of mesh variational inequalities.

2. Iterative solution of  mesh variational  inequalities with positive definite matrices.

Stationary one-step iterative methods, convergence, optimal iterative parameters, problems of preconditioning. Relaxation-type methods (Jacoby, Gauss-Seidel and SOR methods).  Splitting iterative methods.
Details of implementation: projection on a convex set and related problems, control of accuracy and stopping criteria.
Examples.

3. Mesh variational inequalities with saddle matrices.

Finite element and/or finite difference approximations of  the variational inequalities with constraints  to gradient of a solution (elasticity-plasticity problem, water filtration problem), and approximation to  a problem of control in right-hand side of an elliptic equation.
Different formulations of the resulting finite dimensional problems.

4.  Iterative solution of  mesh variational  inequalities with saddle matrices.

Stationary one-step iterative methods (including Arrow-Hurvitz  and Uzawa-type methods). Splitting iterative methods.
Examples.

5. Remarks and generalizations.

Domain decomposition methods and  multigrid methods for the variational inequalities.
Regularization and penalty approach and  Newton-type methods  for solving variational inequalities.

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More information: Erkki Laitinen

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