A Correction Function-based KFBI Method for Brinkman Interface Problems

TL;DR

This paper introduces a novel correction-function-based kernel-free boundary integral method for accurately solving interface problems with discontinuous coefficients, demonstrated through numerical experiments.

Contribution

It develops a new CF-KFBI approach that recasts interface problems as boundary integral equations and employs local correction functions for improved accuracy.

Findings

The method achieves high accuracy in fixed- and moving-interface problems.

Numerical experiments validate the efficiency and robustness of the approach.

Abstract

In this work, we propose a correction-function-based kernel-free boundary integral (CF-KFBI) method for solving Stokes- and Brinkman-type interface problems. We begin by recasting the original interface problem with discontinuous coefficients as boundary integral equations, in which the integral operators can be interpreted as boundary data for potential functions that satisfy simpler interface problems without coefficient discontinuities. Each such interface problem is discretized using a corrected Marker-and-Cell (MAC) scheme. Within a narrow band around the interface, we introduce a local correction function that represents the solution jump, leading to a local Cauchy problem. This problem is solved with a collocation method, for which we provide criteria for a minimal choice of collocation points and prove solvability. Several numerical experiments, including both fixed- and moving-interface problems, are presented to demonstrate the accuracy and efficiency of the proposed method.