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 point 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 M-matrix property. Consequently, for the elliptic problem without interface (i.e., $\Gamma$ is empty), our compact FDM satisfies the discrete maximum principle, which guarantees the theoretical sixth-order convergence. We also derive sixth-order compact (4-point for corners and 6-point for edges) FDMs having the M-matrix property at any boundary point subject to (mixed) Dirichlet/Neumann/Robin boundary conditions. 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 proposed FDMs are independent of the choice representing $\Gamma$ and are also applicable if the jump conditions on $\Gamma$ only depend on the geometry (e.g., curvature) of the curve $\Gamma$. 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$-\nabla \cdot( a\nabla u)=f$（定义域$\Omega\backslash \Gamma$，其中$\Gamma$为$\Omega$内的光滑界面）发展了六阶混合有限差分方法（FDMs）。变量标量系数$a>0$与源项$f$可能在$\Gamma$两侧存在间断性。混合FDM在网格内部正则点采用9点紧致模板，在$\Gamma$附近不规则点采用13点模板。对于远离$\Gamma$的内部正则点，我们构造了满足M矩阵性质的六阶9点紧致FDM。因此，针对无界面椭圆问题（即$\Gamma$为空集），该紧致FDM满足离散最大值原理，保证了理论六阶收敛性。我们进一步推导了在任意混合Dirichlet/Neumann/Robin边界条件边界点处具有M矩阵性质的六阶紧致FDM（角点采用4点模板，边点采用6点模板）。针对$\Gamma$附近的不规则点，我们提出采用五阶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，本文提出的高阶FDM依赖于系数$a$、源项$f$、界面曲线$\Gamma$、沿$\Gamma$的两个跳跃函数以及$\partial\Omega$边界函数的高阶（偏）导数。数值计算中，始终通过函数值逼近所有所需高阶（偏）导数而不损失精度。所提出的FDM与$\Gamma$的表示形式无关，且适用于跳跃条件仅依赖于曲线$\Gamma$几何特征（如曲率）的情形。数值实验证实，本文提出的混合FDM在$l_{\infty}$范数下对椭圆界面问题具有六阶收敛性。