φ-FD: A Well-conditioned Finite Difference Method Inspired by φ-FEM for General Geometries on Elliptic PDEs

• Published in: Journal of scientific computing Vol. 104; no. 1; p. 23
• Main Authors: Duprez, Michel, Lleras, Vanessa, Lozinski, Alexei, Vigon, Vincent, Vuillemot, Killian
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.07.2025; Springer Nature B.V; Springer Verlag
• Subjects: Algorithms; Analysis of PDEs; Boundary conditions; Computational Mathematics and Numerical Analysis; Elliptic differential equations; Elliptic functions; Finite difference method; Finite element analysis; Finite element method; Fourier transforms; Mathematical analysis; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Numerical Analysis; Parabolic differential equations; Partial differential equations; Python; Theorems; Theoretical; Partial differential equations; Unfitted method; 65N06; General geometries; 74S20; 65N55; Finite difference method; general geometry
• ISSN: 0885-7474; 1573-7691
• DOI: 10.1007/s10915-025-02914-0

This paper presents a new finite difference method, called φ -FD, inspired by the φ -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 the 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 φ -FD method compared to standard finite element methods and the Shortley-Weller approach.