In this paper we develop finite difference schemes for elliptic problems with piecewise continuous coefficients that have (possibly huge) jumps across fixed internal interfaces. In contrast with such problems involving one smooth non-intersecting interface, that have been extensively studied, there are very few papers addressing elliptic interface problems with intersecting interfaces of coefficient jumps. It is well known that if the values of the permeability in the four subregions around a point of intersection of two such internal interfaces are all different, the solution has a point singularity that significantly affects the accuracy of the approximation in the vicinity of the intersection point. In the present paper we propose a fourth-order 9-point finite difference scheme on uniform Cartesian meshes for an elliptic problem whose coefficient is piecewise constant in four rectangular subdomains of the overall two-dimensional rectangular domain. Moreover, for the special case when the intersecting point of the two lines of coefficient jumps is a grid point, such a compact scheme, involving relatively simple formulas for computation of the stencil coefficients, can even reach sixth order of accuracy. Furthermore, we show that the resulting linear system for the special case has an M-matrix, and prove the theoretical sixth order convergence rate using the discrete maximum principle. Our numerical experiments demonstrate the fourth (for the general case) and sixth (for the special case) accuracy orders of the proposed schemes. In the general case, we derive a compact third-order finite difference scheme, also yielding a linear system with an M-matrix. In addition, using the discrete maximum principle, we prove the third order convergence rate of the scheme for the general elliptic cross-interface problem.

翻译：本文针对系数在固定内部界面上存在（可能巨大）跳跃的分段连续系数椭圆问题，构建了有限差分格式。与仅含一条光滑不相交界面的此类问题（已有广泛研究）不同，涉及系数跳跃界面相交的椭圆界面问题鲜有文献探讨。众所周知，当两条此类内部界面交点周围四个子区域的渗透率值均不相同时，解会在交点处出现点奇异性，显著影响该点附近近似解的精度。本文提出一种基于均匀笛卡尔网格的9点四阶有限差分格式，用于求解系数在二维矩形区域内的四个矩形子域中分段常数的椭圆问题。此外，在系数跳跃的两条直线交点为网格点的特殊情形下，该紧凑格式（其模板系数计算采用相对简单的公式）甚至能达到六阶精度。进一步，我们证明了此特殊情形下生成的线性系统具有M矩阵性质，并利用离散极值原理证明了理论六阶收敛速度。数值实验展示了所提格式在一般情形下达到四阶精度，在特殊情形下达到六阶精度。针对一般情形，我们推导了一种紧凑的三阶有限差分格式，其线性系统同样具有M矩阵性质，并基于离散极值原理证明了该格式在一般椭圆交叉界面问题中的三阶收敛速度。