A correction function-based kernel-free boundary integral method for elliptic PDEs with implicitly defined interfaces

TL;DR

This paper introduces a correction function-based kernel-free boundary integral method for elliptic PDEs with complex interfaces, simplifying implementation and improving accuracy for challenging boundary problems.

Contribution

It presents a novel correction function approach within a kernel-free boundary integral framework, enabling efficient and accurate solutions for elliptic PDEs with implicitly defined interfaces.

Findings

Method is accurate for high-contrast coefficients

Efficient for problems with close interfaces

Easy to implement in 3D applications

Abstract

This work addresses a novel version of the kernel-free boundary integral (KFBI) method for solving elliptic PDEs with implicitly defined irregular boundaries and interfaces. We focus on boundary value problems and interface problems, which are reformulated into boundary integral equations and solved with the matrix-free GMRES method. In the KFBI method, evaluating boundary and volume integrals only requires solving equivalent but much simpler interface problems in a bounding box, for which fast solvers such as FFTs and geometric multigrid methods are applicable. For the simple interface problem, a correction function is introduced for both the evaluation of right-hand side correction terms and the interpolation of a non-smooth potential function. A mesh-free collocation method is proposed to compute the correction function near the interface. The new method avoids complicated derivation for derivative jumps of the solution and is easy to implement, especially for the fourth-order method in three space dimensions. Various numerical examples are presented, including challenging cases such as high-contrast coefficients, arbitrarily close interfaces and heterogeneous interface problems. The reported numerical results verify that the proposed method is both accurate and efficient.