Convergent Sixth-order Compact Finite Difference Method for Variable-Coefficient Elliptic PDEs in Curved Domains

Abstract: Finite difference methods (FDMs) are widely used for solving partial differential equations (PDEs) due to their relatively simple implementation. However, they face significant challenges when applied to non-rectangular domains and in establishing theoretical convergence, particularly for high-order schemes. In this paper, we focus on solving the elliptic equation $-\nabla \cdot (a\nabla u)=f$ in a two-dimensional curved domain $\Omega$, where the diffusion coefficient $a$ is variable and smooth. We propose a sixth-order $9$-point compact FDM that only utilizes the grid points in $(h \mathbb{Z}^2)\cap \Omega$ for any mesh size $h>0$, without relying on ghost points or information outside $\overline{\Omega}$. All the boundary stencils near $\partial \Omega$ have at most $6$ different configurations and use at most $8$ grid points inside $\Omega$. We rigorously establish the sixth-order convergence of the numerically approximated solution $u_h$ in the $\infty$-norm. Additionally, we derive a gradient approximation $\nabla u$ directly from $u_h$ without solving auxiliary equations. This gradient approximation achieves proven accuracy of order $5+\frac{1}{q}$ in the $q$-norm for all $1\le q\le \infty$ (with a logarithmic factor $\log h$ for $1\le q<2$). To validate our proposed sixth-order compact finite different method, we provide several numerical examples that illustrate the sixth-order accuracy and computational efficiency of both the numerical solution and the gradient approximation for solving elliptic PDEs in curved domains.