Efficient solvers for discretized elliptic vector-valued problems - Martikainen, Janne (2002)
In this thesis, some aspects of efficient methods for solving discretized, vector-valued partial differential equations (PDEs) are studied. The purpose is to develop new techniques and show how old ideas can be extended to a larger class of problems. However, parallelization of the algorithms and parallel computing have been excluded of the scope of this thesis. This thesis consists of an introduction and seven articles. It concentrates on several issues involved in the solution of elliptic PDEs. The combining factor of the included articles is the use of fast direct solvers and various multilevel techniques as preconditioners in the iterative solution of elliptic, especially vector-valued, PDEs. A new fast direct solver is developed for elliptic problems with divergence constraints. It uses finite element approximation with a stable basis function combination. A few multilevel techniques for approximating Schur complements or preconditioning boundary Lagrange multipliers are studied. They give simple and efficient approximations also for three-dimensional problems. The use of fast solvers for systems of equations to eigenvalue problems is considered. It is shown that under suitable conditions, fixed accuracy estimates for a few smallest or largest eigenvalues can be achieved by a number of operations, which depends linearly on the size of the linear system. A spectrally optimal block diagonal preconditioner with more than two blocks for the mixed formulation of the linear elasticity problem is constructed and tested. Finally, there are two application articles. In the first of them, an algebraic multigrid method is used for the numerical simulation of ultrasonic transducers. Both resonance and transient analysis of the transducers are performed. In the second one, the fast direct solver developed earlier in this thesis is combined with a fictitious domain method to obtain a generalized Stokes solver for more complicated geometries. The generalized Stokes solver is applied further to the time-dependent Navier-Stokes equations managed with an operator splitting technique.