A comparison of the Coco-Russo scheme and $\protect\mathghost$-FEM for elliptic equations in arbitrary domains

TL;DR

This paper compares the Coco-Russo finite-difference scheme and the $ ext{ extbackslash mathghost}$-FEM finite-element method for solving the Poisson equation in complex domains, highlighting their relative performance and analytical properties.

Contribution

It provides a detailed comparison between two numerical schemes for elliptic equations in arbitrary domains, combining analytical insights and numerical experiments.

Findings

Coco-Russo scheme performs well in certain geometries

$ ext{ extbackslash mathghost}$-FEM shows advantages in boundary condition handling

Analytical results support numerical observations

Abstract

In this paper, a comparative study between the Coco-Russo scheme (based on finite-difference scheme) and the $\mathghost$-FEM (based on finite-element method) is presented when solving the Poisson equation in arbitrary domains. The comparison between the two numerical methods is carried out by presenting analytical results from the literature \cite{cocoStissi,astuto2024nodal}, together with numerical tests in various geometries and boundary conditions.