{\varphi}-FD : A well-conditioned finite difference method inspired by {\varphi}-FEM for general geometries on elliptic PDEs

TL;DR

This paper introduces {}-FD, a finite difference method inspired by {}-FEM, that efficiently solves elliptic PDEs on complex geometries using Cartesian grids, with proven convergence and well-conditioned matrices.

Contribution

The paper proposes a novel {}-FD method that combines the simplicity of Cartesian grids with well-conditioned matrices for elliptic PDEs on complex geometries.

Findings

Proves quasi-optimal convergence rates in multiple norms.

Demonstrates the matrix is well-conditioned.

Numerical experiments show competitive performance with FEM and Shortley-Weller methods.

Abstract

This paper presents a new finite difference method, called {\varphi}-FD, inspired by the {\phi}-FEM approach for solving elliptic partial differential equations (PDEs) on general geometries. The proposed method uses Cartesian grids, ensuring simplicity in implementation. Moreover, contrary to the previous finite difference scheme on non-rectangular domain, the associated matrix is well-conditioned. The use of a level-set function for the geometry description makes this approach relatively flexible. We prove the quasi-optimal convergence rates in several norms and the fact that the matrix is well-conditioned. Additionally, the paper explores the use of multigrid techniques to further accelerate the computation. Finally, numerical experiments in both 2D and 3D validate the performance of the {\varphi}-FD method compared to standard finite element methods and the Shortley-Weller approach.