In this paper, we develop sixth-order hybrid finite difference methods (FDMs) for the elliptic interface problem $-\nabla \cdot( a\nabla u)=f$ in $\Omega\backslash \Gamma$, where $\Gamma$ is a smooth interface inside $\Omega$. The variable scalar coefficient $a>0$ and source $f$ are possibly discontinuous across $\Gamma$. The hybrid FDMs utilize a $9$-point compact stencil at any interior regular points of the grid and a $13$-point stencil at irregular points near $\Gamma$. For interior regular points away from $\Gamma$, we obtain a sixth-order $9$-point compact FDM satisfying the sign and sum conditions for ensuring the M-matrix property. We also derive sixth-order compact ($4$-point for corners and $6$-point for edges) FDMs satisfying the sign and sum conditions for the M-matrix property at any boundary point subject to (mixed) Dirichlet/Neumann/Robin boundary conditions. Thus, for the elliptic problem without interface (i.e., $\Gamma$ is empty), our compact FDM has the M-matrix property for any mesh size $h>0$ and consequently, satisfies the discrete maximum principle, which guarantees the theoretical sixth-order convergence. For irregular points near $\Gamma$, we propose fifth-order $13$-point FDMs, whose stencil coefficients can be effectively calculated by recursively solving several small linear systems. Theoretically, the proposed high order FDMs use high order (partial) derivatives of the coefficient $a$, the source term $f$, the interface curve $\Gamma$, the two jump functions along $\Gamma$, and the functions on $\partial \Omega$. Numerically, we always use function values to approximate all required high order (partial) derivatives in our hybrid FDMs without losing accuracy. Our numerical experiments confirm the sixth-order convergence in the $l_{\infty}$ norm of the proposed hybrid FDMs for the elliptic interface problem.

翻译：本文针对椭圆界面问题 $-\nabla \cdot( a\nabla u)=f$（$\Omega\backslash \Gamma$ 区域），其中 $\Gamma$ 是 $\Omega$ 内的光滑界面，发展了六阶混合有限差分方法（FDMs）。标量变系数 $a>0$ 和源项 $f$ 可能跨越 $\Gamma$ 间断。该混合FDMs在网格内部规则点采用9点紧致模板，在 $\Gamma$ 附近不规则点采用13点模板。对于远离 $\Gamma$ 的内部规则点，我们构造了满足符号和条件（确保M矩阵性质）的六阶9点紧致FDM。同时，针对任意受（混合）Dirichlet/Neumann/Robin边界条件的边界点，推导了满足M矩阵性质符号和条件的六阶紧致FDM（角点4点、边点6点）。因此，对于无界面椭圆问题（即 $\Gamma$ 为空集），我们的紧致FDM对任意网格尺寸 $h>0$ 具有M矩阵性质，进而满足离散最大值原理，保证理论六阶收敛性。对于 $\Gamma$ 附近的不规则点，我们提出五阶13点FDM，其模板系数可通过递归求解若干小型线性方程组高效计算。理论上，所提高阶FDM利用了系数 $a$、源项 $f$、界面曲线 $\Gamma$、沿 $\Gamma$ 的两个跳跃函数以及 $\partial \Omega$ 上函数的高阶（偏）导数。数值实现中，我们始终使用函数值近似所有所需高阶（偏）导数而不损失精度。数值实验证实了所提混合FDM在 $l_{\infty}$ 范数下对椭圆界面问题的六阶收敛性。