Publication Date

Summer 6-2-2022

Abstract

Within recent decades, spectral methods have become an important technique in numerical computing for solving partial differential equations. This is due to their superior accuracy when compared to finite difference and finite element methods. For such spectral approximations, the convergence rate is solely dependent on the smoothness of the solution yielding the potential to achieve spectral accuracy. We present an iterative approach for solving the two-dimensional Helmholtz problem posed on a rectangular domain subject to Dirichlet boundary conditions that is well-conditioned, low in memory, and of sub-quadratic complexity. The proposed approach spectrally approximates the partial differential equation by means of modal Chebyshev integration matrices. Implementation of the boundary conditions is achieved through a technique known as ``integration preconditioning," although we refer to the technique as integration sparsification. The spectral method presented represents certain partial differential operators in terms of sparse, banded integration matrices. In this work, there are $N+1$ Chebyshev modes associated with each coordinate direction. Therefore, there are $n = (N + 1)^2$ modes in total. For the truncations considered, our method empirically yields a linear set-up cost, followed by a sub-quadratic solve complexity of $\mathcal{O}(n^{1.6})$.

Degree Name

Mathematics

Level of Degree

Masters

Department Name

Mathematics & Statistics

First Committee Member (Chair)

Stephen Lau

Second Committee Member

Deborah Sulsky

Third Committee Member

Jacob Bayer Schroder

Language

English

Keywords

Spectral-Tau Method, Helmholtz Equation, Numerical Methods, Partial Differential Equations, Integration Sparsification, Preconditioning, Iterative Methods, Chebyshev Polynomials

Document Type

Thesis

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