A diffuse interface box method for elliptic problems

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.