A kernel-free boundary integral method for implicitly defined surfaces

Abstract

The kernel-free boundary integral (KFBI) method is a structured grid method for general elliptic partial differential equations. Unlike the standard boundary integral method, it avoids direct evaluation of volume and boundary integrals, which needs to know analytical expressions for the integral kernels. To evaluate a boundary or volume integral, the KFBI method first solves a corrected interface problem on a structured grid and then the numerical solution on the structured grid is interpolated to get approximate values of the integral at points on the boundary. Selection of control points of the boundary plays a key role in the KFBI method since both the correction for the interface equations and the interpolation with the structured grid based solution involve calculation of tangential derivatives of boundary data while stability and efficiency of the numerical differentiation critically depend on the distribution of control points. This work proposes a new point selection method, based on an overlapping surface decomposition of the boundary, which is implicitly defined by a level set function. The points selected are intersection points of the boundary with the grid lines of an underlying Cartesian grid. By the method, the interpolation stencils can be easily chosen to be locally uniform along a coordinate axis in two space dimensions and locally uniform on a coordinate plane in three space dimensions, which allows efficient numerical differentiation and boundary reconstruction/representation. An additional equilibrating process of boundary data further guarantees stable numerical differentiation. Numerical results demonstrating the method with examples in both two and three space dimensions are presented.