PhD Defense: Fast and Accurate Boundary Element Methods in Three

The Laplace and Helmholtz equations in three dimensions are two of the most important, perhaps the most important, partial differential equations (PDEs) in science, and govern problems in a large number of disciplines, including electromagnetism, acoustics, astrophysics, molecular dynamics, and aerodynamics, among others. Therefore, having fast and accurate numerical solvers for these problems is important.

The boundary element method (BEM) is a powerful method for solving these PDEs. The BEM reduces the dimensionality of the problem by one, allows for the treatment of complex boundary shapes, and treats thin surfaces and multi-domain problems well. The BEM also suffers from a few problems. The entries in the system matrices require the computation of certain boundary integrals, which can be difficult to do accurately, especially in the Galerkin formulation. These matrices are also dense, and when solved conventionally via direct matrix decompositions, require

O(_N_2) storage and run in O(_N_3), where _N_ is the number of discretization unknowns. This can effectively restrict the size of a problem. This dissertation addresses these issues by making three contributions.

First, novel methods inspired by the fast multipole method (FMM) are presented for computing all the boundary integrals that arise in the Galerkin formulation to any accuracy. Integrals involving completely geometrically separated triangles are non-singular, and are computed using a technique based on spherical harmonics and multipole expansions and translations, which require the integration of polynomial functions over the triangles. Integrals involving cases where the triangles have common vertices, edges, or are coincident are treated via scaling and symmetry arguments, combined with automatic recursive geometric decomposition of the integrals.

Second, the FMM is used to accelerate the BEM. The FMM is usually designed around monopole and dipole sources, not the integral expressions in the BEM. To apply the FMM to these expressions, the internal logic of the FMM must be changed, but this can be difficult to do. The correction factor matrix method is presented, which works by approximating the integrals using a quadrature. The quadrature points are treated as monopole and dipole sources, which are plugged directly into current FMM codes. Any inaccuracies from the quadrature are corrected during a correction factor step. This method reduces the quadratic and cubic scalings of the BEM to linear.

Third, computational software is developed for calculating the solutions to acoustic scattering problems involving spheroids and disks. This software is used to verify the accuracy of the BEM for the Helmholtz equation. This software uses spheroidal wave functions to analytically build the solutions to these problems. However, the spheroidal wave functions are notoriously difficult to compute. Methods for computing these special functions accurately are given.

The product of these three contributions is a fast and accurate BEM solver for the Laplace and Helmholtz equations. Where possible, the companion code for this dissertation has been released online and is available for download.

Examining Committee:

Chair: Dr. Ramani Duraiswami

Dean’s rep: Dr. Thomas Antonsen

Members: Dr. Nail Gumerov

Dr. Howard Elman

Dr. Amitabh Varshney