Error estimates on a finite volume method for diffusion problems with interface on Eulerian grids

TL;DR

This paper introduces a vertex-centered MACH-like finite volume method for diffusion problems with interfaces on Eulerian grids, providing error estimates and confirming accuracy through numerical experiments.

Contribution

It develops a new finite volume scheme for interface diffusion problems and derives optimal error estimates, advancing numerical analysis in this area.

Findings

The scheme achieves an asymptotic error of O(h^2 |ln h|) in maximum norm.

Error estimates are rigorously derived for the proposed method.

Numerical experiments confirm the theoretical error bounds.

Abstract

The finite volume methods are frequently employed in the discretization of diffusion problems with interface. In this paper, we firstly present a vertex-centered MACH-like finite volume method for solving stationary diffusion problems with strong discontinuity and multiple material cells on the Eulerian grids. This method is motivated by Frese [No. AMRC-R-874, Mission Research Corp., Albuquerque, NM, 1987]. Then, the local truncation error and global error estimates of the degenerate five-point MACH-like scheme are derived by introducing some new techniques. Especially under some assumptions, we prove that this scheme can reach the asymptotic optimal error estimate $O(h^2 |\ln h|)$ in the maximum norm. Finally, numerical experiments verify theoretical results.