Fast multipole boundary element solutions with inexact Krylov iterations and relaxation strategies

Abstract

Boundary element methods (BEM) have been used for years to solve a multitude of engineering problems, ranging from Bioelectrostatics, to fluid flows over micro-electromechanical devices and deformations of cell membranes. Only requiring the discretization of a surface into panels rather than the entire domain, they effectively reduce the dimensionality of a problem by one. This reduction in dimensionality nevertheless comes at a cost. BEM requires the solution of a large, dense linear system with each matrix element formed of an integral between two panels, often performed used an iterative solver based on Krylov subspace methods. This requires the repeated calculation of a matrix vector product that can be approximated using a hierarchical approximation known as the fast multipole method (FMM). While adding complexity, this reduces order of the time-to-solution from O(cN^2) to OcN), where c is some function of the condition number of the dense matrix. This thesis obtains algorithmic speedups for the solutions of FMM-BEM systems by applying the mathematical theory behind inexact matrix-vector products to our solver, implementing a relaxation scheme to control the error incurred by the FMM in order to minimize the total time-to-solution. The theory is extensively verified for both Laplace equation and Stokes flow problems, with an investigation to determine how further problems may benefit from the addition of a relaxed solver. We also present experiments for the Stokes flow around both single and multiple red blood cells, an area of ongoing research, showing good speedups that would be applicable for any other code that chose to implement a similar relaxed solver. All of these results are obtained with an easy-to-use, extensible and open-source FMM-BEM code.