A kernel-free boundary integral method for elliptic interface problems on surfaces

TL;DR

This paper introduces a kernel-free boundary integral method for elliptic PDEs on surfaces that avoids explicit kernel functions, using interpolation and multigrid solvers to achieve efficient and accurate solutions for boundary and interface problems.

Contribution

It develops a second-order kernel-free boundary integral method for elliptic equations on surfaces, simplifying implementation and improving computational efficiency.

Findings

Method achieves second-order accuracy.

Numerical experiments confirm efficiency and robustness.

Applicable to boundary value and interface problems.

Abstract

This work presents a generalized boundary integral method for elliptic equations on surfaces, encompassing both boundary value and interface problems. The method is kernel-free, implying that the explicit analytical expression of the kernel function is not required when solving the boundary integral equations. The numerical integration of single- and double-layer potentials or volume integrals at the boundary is replaced by interpolation of the solution to an equivalent interface problem, which is then solved using a fast multigrid solver on Cartesian grids. This paper provides detailed implementation of the second-order version of the kernel-free boundary integral method for elliptic PDEs defined on an embedding surface in $\mathbb{R}^3$ and presents numerical experiments to demonstrate the efficiency and accuracy of the method for both boundary value and interface problems.