Multigrid preconditioning of the linear elasticity equation

Abstract

Multigrid methods are known to be very efficient linear solvers for 2nd order elliptic PDEs. In this thesis we consider multigrid methods, and the use of such as preconditioners, when solving the linear elasticity equation. An introduction to modeling elasticity is given and two discretization techniques; the finite-element and the virtual-element method are presented. The conjugate gradient method is described and a result relating the convergence rate to the condition number is established. Building blocks of the multigrid method are presented. As a part of the study, an implmentation of the smoothed aggregation algebraic multigrid method due to Vanek et al. has been done for elasticity. Numerical tests on simple problems shows that the convergence rate of our implmentation has a moderate dependence on the problem size when used as a standalone solver. When used as a preconditioner for conjugate gradient the convergence rate is found to be practically independent of the problem size compared to standard preconditioners.