论文标题
紧凑的9点有限差异方法具有高精度顺序和/或M-Matrix属性，用于椭圆形跨距问题问题
Compact 9-Point Finite Difference Methods with High Accuracy Order and/or M-Matrix Property for Elliptic Cross-Interface Problems
论文作者
论文摘要
在本文中，我们开发了有限的差异方案，以解决具有（可能很大的）跨固定内部接口的分段连续系数的椭圆问题。与涉及一个经过广泛研究的一个平滑非交流界面的此类问题相反，很少有论文解决椭圆界面问题的相交系数跳跃界面的问题。众所周知，如果在两个这样的内部接口的交点周围的四个子区域中的渗透率值都不同，则该溶液具有一个点奇异性，从而显着影响交叉点附近近似值的准确性。在本文中，我们针对椭圆形问题的均匀笛卡尔网格上提出了一个四阶9分差差方案，该方案的系数在整个二维矩形域的四个矩形子域中是分段常数。此外，对于特殊情况，当系数跳跃的两行相交点是一个网格点时，这种紧凑的方案（涉及用于计算模板系数的相对简单公式）甚至可以达到准确性的第六顺序。此外，我们表明，特殊情况的最终线性系统具有m-matrix，并使用离散最大原理证明了理论上的第六阶收敛速率。我们的数值实验证明了所提出的方案的第四个（通常情况）和第六（特殊情况）的精度顺序。在一般情况下，我们得出了一个紧凑的三阶有限差方案，还产生了带有M-Matrix的线性系统。此外，使用离散的最大原理，我们证明了一般椭圆形跨接口问题方案的三阶收敛率。
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.
