Utilization of Discrete Mathematics in Parallel Boundary Element Method

Abstract

This thesis describes techniques used to implement the boundary element method for the Helmholtz transmission problem in distributed compute environment. These techniques are based on principles from graph theory and are applicable to matrix assembly and matrix-vector product when the matrices are represented in a block form.

Firstly, we formulate the Helmholtz transmission problem in three dimensions and recall the appropriate fundamental theory used by the relaxed local multi-trace formulation, which is our method of choice to solve the problem. The relaxed local multi-trace formulation is then devised for general problem settings, that is we consider a finite number of domains, each with its own wave number, together with transmission conditions to enforce continuity of the solution across boundaries of the domains.

Afterwards, we introduce the reader to fast boundary element method which is used to approximate the corresponding full matrices. To this end, we resort to using the algebraic version of the adaptive cross approximation technique and discuss its advantages and also its insufficiencies. We propose an improved adaptive cross approximation algorithm which aims to solve the discussed issues and we demonstrate its superiority on a variety of interior Hemlholtz problems.

Furthermore, we introduce a method of workload assignment for distributed parallel matrix assembly and matrix-vector product and argue its optimality with respect to the number of degrees of freedom required by each processing unit.

Finally, we provide the results of numerical experiments which analyze the effects a block representation has on parallel efficiency of matrix assembly and matrix-vector product. In relation to the relaxed local multi-trace formulation, we argue that additional division of processing power might be required and formulate the appropriate algorithms. In the end, we offer the results of parallel experiments on a selection of Helmholtz transmission problems both with and without junction (cross) points and measure how the number of GMRES iterations is affected by the relaxation parameter of the relaxed local multi-trace formulation.