Space-Time Parallel Algorithms for Boundary Element Methods

Abstract

In this thesis we revisit theoretical background for space­time boundary element methods for the heat equation and its implementation. We restrict ourselves to solving the one­, and two dimensional Dirichlet heat equation. A new approach is proposed to approximate the Galerkin matrix entries in a semianalytical fashion, requiring a reduced order of quadrature. This method can be applied to non­uniform meshes, but is restricted to right­triangular meshes. Using this approach, a system of Galerkin equations is created and solved iteratively with the use of the Generalised Residual Method (GMRES). Operator preconditioners and preconditioners originating from domain decomposition methods are summarised and implemented for a two dimensional Dirichlet problem. In the case of operator preconditioning, a diagonal duality pairing proposed by Stevenson and van Venetië [23] is used in the implementations. The Restricted Additive Schwarz method is considered as both a preconditioner and a basic iterative method. The Calderón preconditioner, an operator preconditioner, is used to test the efficiency of parallel preconditioned GMRES implementations, as this preconditioner provides a dense matrix. For different amount of processes, the parallel GMRES implementations are investigated. Using row­wise decomposition, parallel GMRES becomes increasingly time­efficient, as the level of refinement increases. However, the Induced Dimension Reduction method, a different non­symmetric solver, currently outperforms the parallel GMRES implementation.