A high-order integral algorithm for highly singular PDE solutions in Lipschitz domains

Abstract

We present a new algorithm, based on integral equation formulations, for the solution of constant-coefficient elliptic partial differential equations (PDE) in closed two-dimensional domains with non-smooth boundaries; we focus on cases in which the integral-equation solutions as well as physically meaningful quantities (such as, stresses, electric/magnetic fields, etc.) tend to infinity at singular boundary points (corners). While, for simplicity, we restrict our discussion to integral equations associated with the Neumann problem for the Laplace equation, the proposed methodology applies to integral equations arising from other types of PDEs, including the Helmholtz, Maxwell, and linear elasticity equations. Our numerical results demonstrate excellent convergence as discretizations are refined, even around singular points at which solutions tend to infinity. We demonstrate the efficacy of this algorithm through applications to solution of Neumann problems for the Laplace operator over a variety of domains—including domains containing extremely sharp concave and convex corners, with angles as small as π/100 and as large as 199π/100.