Matrix equation solving of PDEs in polygonal domains using conformal mappings

Abstract

Abstract We explore algebraic strategies for numerically solving linear elliptic partial differential equations in polygonal domains. To discretize the polygon by means of structured meshes, we employ Schwarz–Christoffel conformal mappings, leading to a multiterm linear equation possibly including Hadamard products of some of the terms. This new algebraic formulation allows us to clearly distinguish between the role of the discretized operators and that of the domain meshing. Various algebraic strategies are discussed for the solution of the resulting matrix equation.