A fourth-order kernel-free boundary integral method for implicitly defined surfaces in three space dimensions

Abstract

The kernel-free boundary integral (KFBI) method is a finite difference version of the traditional boundary integral method for elliptic and parabolic partial differential equations on complex domains. It evaluates boundary or volume integrals involved in the solution of boundary integral equations (BIEs) by solving equivalent but simple interface problems on regular grids, so that the integral kernel or Green function is never needed or computed. This is the essential difference of the KFBI method from the traditional ones. It takes advantage of the well-conditioning property of discrete BIEs so that the number of Krylov subspace iterations is essentially independent of discretization parameter or system dimension. This paper presents a fourth-order kernel-free boundary integral method for second-order elliptic partial differential equations on complex domains in three space dimensions, whose boundaries are given by implicitly defined surfaces. It represents the domain boundary and discretizes data on it by intersection points of the surface with Cartesian grid lines. The approach has a variety of advantages. The current work solves simple interface problems with corrected 27-point compact finite difference schemes and calculates the discrete equations with a fast Fourier transform based elliptic solver. Numerical examples show that the proposed method is efficient as well as accurate.