Toward Simulating 2D Cell Surfaces in a Disk

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