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 \(1\leqslant q\leqslant \infty \) (with a logarithmic factor \(\log h\) for \(1\leqslant q<2\) ). 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 approximate gradient for solving elliptic PDEs in curved domains.
