Laplace's equation on perturbed domains

Abstract

Laplace's equation is a prototypical elliptic PDE that appears in many electromagnetic and fluid dynamics problems. We develop two methods for solving Laplace's equation on domains that are perturbations of a circle. These methods are derived from governing equations and applied to several test cases. Both Dirichlet and Neumann boundary conditions are considered. We verify our methods by constructing exact solutions for the perturbed geometries.