Date of Award

Spring 1-1-2011

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Applied Mathematics

First Advisor

Per-Gunnar Martinsson

Second Advisor

Gregory Beylkin

Third Advisor

Bradley Alpert

Abstract

The dissertation describes fast, robust, and highly accurate numerical methods for solving boundary value problems associated with elliptic PDEs such as Laplace's and Helmholtz' equations, the equations of elasticity, and time-harmonic Maxwell's equation. In many areas of science and engineering, the cost of solving such problems determines what can and cannot be modeled computationally. Elliptic boundary value problems may be solved either via discretization of the PDE (e.g., finite element methods) or by first reformulating the equation as an integral equation, and then discretizing the integral equation. In either case, one is left with the task of solving a system of linear algebraic equations that could be very large. There exist a broad range of schemes with linear complexity for solving these equations (multigrid, preconditioned Krylov methods, etc). Most of these schemes are based on ``iterative'' techniques that build a sequence of approximate solutions that converges to the exact solution. In contrast, the methods described here are ``direct'' in the sense that they construct an approximation to the inverse (or LU/Cholesky factorization) of the coefficient matrix. Such direct solvers tend to be more robust, versatile, and stable than iterative methods, but have until recently been considered prohibitively expensive for large scale problems. The objective of the dissertation is to demonstrate that in important environments it is possible to construct an approximate inverse with linear computational cost. The methods are for a single solve competitive with the best iterative methods, and can be far faster than any previously available methods in situations where the same coefficient matrix is used in a sequence of problems. In addition, a new discretization technique for elliptic boundary value problems is proposed. The idea is to first compute the solution operator of a large collection of small domains. The small domains are chosen such that the operator is easily computed to high accuracy. A global equilibrium equation is then built by equating the fluxes through all internal domain boundaries. The resulting linear system is well-suited to the newly developed fast direct solvers.

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