Publication Date

Spring 7-29-2025

Abstract

Certain evolution models of cell surfaces (treated in two-dimensions) involve the solution of the Helmholtz equation with jump conditions enforced on an immersed closed curve. This thesis presents a sparse, modal spectral method for solving such Helmholtz problems. The solution is required to be continuous across the curve, but with a jump discontinuity in the normal derivative proportional to the planar curvature. The method relies on classical Fourier-Chebyshev basis functions, with the application of modal Chebyshev integration matrices to achieve sparse, banded approximations of the Helmholtz equation. The method achieves spectral convergence, despite the inherent low regularity of the relevant solutions. While the target application involves a disk domain, to focus on complexity issues and the complication of the jump conditions, this thesis adopts an annular domain. Three separate scenarios are considered: a single annulus, a multi-annulus domain decomposition with disjoint subdomain interiors, and a multi-annulus domain decomposition for which precisely two annuli (subdomains) overlap. The third scenario allows for treatment of the jump conditions with the overlap region containing an immersed curve. For each scenario, this thesis considers the structure of the linear system arising in the corresponding approximation and a strategy for its fast inversion. Numerical tests of the described spectral method examine accuracy and convergence.

Degree Name

Mathematics

Level of Degree

Masters

Department Name

Mathematics & Statistics

First Committee Member (Chair)

Stephen Lau

Second Committee Member

Owen Lewis

Third Committee Member

Anna Nelson

Language

English

Keywords

Helmholtz, annulus, curve, polynomial, Chebyshev, Fourier

Document Type

Thesis

Comments

In this thesis, the following is described: Chebyshev polynomials, the Helmholtz problem on an annulus, the multi-annulus Helmholtz problem, and the multi-annulus with overlap Helmholtz problem.

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