A diffuse interface box method for elliptic problems

TL;DR

This paper introduces a diffuse interface box method (DIBM) for solving elliptic problems with Dirichlet boundary conditions on complex geometries, providing error estimates and numerical validation.

Contribution

The paper presents a novel diffuse interface box method (DIBM) for elliptic problems, with theoretical error analysis and numerical validation on complex geometries.

Findings

Error estimates in $H^1$ and $L^2$ norms are derived.

Numerical results confirm the theoretical error bounds.

The method effectively handles complex geometries.

Abstract

We introduce a diffuse interface box method (DIBM) for the numerical approximation on complex geometries of elliptic problems with Dirichlet boundary conditions. We derive a priori $H^1$ and $L^2$ error estimates highlighting the r\^{o}le of the mesh discretization parameter and of the diffuse interface width. Finally, we present a numerical result assessing the theoretical findings.