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

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 on uniform Cartesian grids within the domain, not 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 $1 \leq q \leq \infty$ (with a logarithmic factor $\log h$ for $1 \leq q < 2$). 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.