Finite Volume Formulation of the MIB Method for Elliptic Interface Problems

TL;DR

This paper introduces a novel finite volume formulation of the MIB method that combines the strengths of both approaches to accurately solve elliptic interface problems with complex geometries, even with lower solution regularity.

Contribution

The paper develops a combined MIB-FVM approach on Cartesian meshes that achieves second order accuracy for complex elliptic interface problems in 2D and 3D.

Findings

Achieves second order convergence in $L_{ ext{infty}}$ and $L_2$ norms.

Validates effectiveness through extensive numerical experiments.

Handles complex interface geometries with high accuracy.

Abstract

The matched interface and boundary (MIB) method has a proven ability for delivering the second order accuracy in handling elliptic interface problems with arbitrarily complex interface geometries. However, its collocation formulation requires relatively high solution regularity. Finite volume method (FVM) has its merit in dealing with conservation law problems and its integral formulation works well with relatively low solution regularity. We propose an MIB-FVM to take the advantages of both MIB and FVM for solving elliptic interface problems. We construct the proposed method on Cartesian meshes with vertex-centered control volumes. A large number of numerical experiments are designed to validate the present method in both two dimensional (2D) and three dimensional (3D) domains. It is found that the proposed MIB-FVM achieves the second order convergence for elliptic interface problems with complex interface geometries in both $L_{\infty}$ and $L_2$ norms.