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

Michel Duprez, Vanessa Lleras, Alexei Lozinski, Vincent Vigon, Killian Vuillemot

TL;DR

This paper introduces φ-FD, a well-conditioned finite-difference method on Cartesian grids for solving the Poisson equation on general geometries described by a level-set φ. The discretization combines a standard discrete Laplacian with boundary-penalization and boundary-near stabilization to enforce Dirichlet conditions, achieving quasi-optimal convergence with an error bound of order about $h^{3/2}$ in combined norms and a matrix condition number bounded by $C h^{-2}$. The method is shown to be compatible with multigrid acceleration and is validated in 2D and 3D against φ-FEM, standard FEM, and Shortley-Weller approaches, where it exhibits supraconvergence in the $H^1$-seminorm and favorable conditioning and timing. An alternative scheme aiming for optimal $H^1$ convergence is proposed, though its convergence proof remains to be established. The work suggests strong potential for efficient, accurate PDE solvers on complex geometries with Cartesian grids, with avenues for extension to Neumann problems, nonlinear PDEs, and neural-operator integrations.

Abstract

This paper presents a new finite difference method, called {\varphi}-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 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.

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

TL;DR

This paper introduces φ-FD, a well-conditioned finite-difference method on Cartesian grids for solving the Poisson equation on general geometries described by a level-set φ. The discretization combines a standard discrete Laplacian with boundary-penalization and boundary-near stabilization to enforce Dirichlet conditions, achieving quasi-optimal convergence with an error bound of order about in combined norms and a matrix condition number bounded by . The method is shown to be compatible with multigrid acceleration and is validated in 2D and 3D against φ-FEM, standard FEM, and Shortley-Weller approaches, where it exhibits supraconvergence in the -seminorm and favorable conditioning and timing. An alternative scheme aiming for optimal convergence is proposed, though its convergence proof remains to be established. The work suggests strong potential for efficient, accurate PDE solvers on complex geometries with Cartesian grids, with avenues for extension to Neumann problems, nonlinear PDEs, and neural-operator integrations.

Abstract

This paper presents a new finite difference method, called {\varphi}-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 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.

Paper Structure

This paper contains 15 sections, 6 theorems, 104 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. General advantages of Cartesian grids
  3. Finite difference method
  4. Finite element method
  5. Article outline
  6. Main results
  7. Link with testFEM
  8. Proof of Theorems \ref{['theo']} and \ref{['theo:cond']}
  9. Alternative scheme
  10. Numerical illustrations
  11. First test case : 2D example
  12. Second test case: a 3D example
  13. Third test case: combination with a multigrid approach
  14. Conclusion and perspectives
  15. Example of code for test-FD in python

Key Result

Theorem 1

Suppose that $\Omega$ is $r$-smooth for a $r>0$ and is defined by a level-set function $\varphi\in\mathcal{C}^2(\overline{\Omega}_h)$ as in eq:phi. Let $u$ be the solution of the continuous system eq:poisson. Suppose that $u\in\mathcal{C}^4(\Omega)$. For $\sigma,~\gamma$ large enough and $h<\dfrac{2

Figures (9)

  • Figure 1: Representation of $\overline{\Omega}_h$, $\Omega_h$ and $\Gamma$.
  • Figure 2: Case $N_j>2$ in the proof of Proposition \ref{['prop:coer']}.
  • Figure 3: Case $N_j=2$ in the proof of Proposition \ref{['prop:coer']}.
  • Figure 4: First test case, a 2D example.$L^{2}$ (left) and $L^{\infty}$ (right) relative errors with respect to the discretization step for $\varphi$-FEM, standard FEM, Shortley-Weller, $\varphi$-FD and $\varphi$-FD2.
  • Figure 5: First test case, a 2D example.$H^1$ relative error (left) and conditioning number (right) with respect to the discretization step for $\varphi$-FEM, standard FEM, Shortley-Weller, $\varphi$-FD and $\varphi$-FD2.
  • ...and 4 more figures

Theorems & Definitions (18)

  • Definition 1
  • Theorem 1: Convergence
  • Remark 1
  • Theorem 2: Conditioning
  • Remark 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Proposition 1: Coercivity
  • ...and 8 more