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arXiv · 2508.16061

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

Pengsong Yin·Wenjun YIng·Yulin Zhang·Han Zhou
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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.

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Pengsong Yin, Wenjun YIng, Yulin Zhang, Han Zhou. 2025-08-22. A kernel-free boundary integral method for elliptic interface problems on surfaces. https://arxiv.org/abs/2508.16061

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