In this paper, we discuss the second-order finite element method (FEM) and finite difference method (FDM) for numerically solving elliptic cross-interface problems characterized by vertical and horizontal straight lines, piecewise constant coefficients, two homogeneous jump conditions, continuous source terms, and Dirichlet boundary conditions. For brevity, we consider a 2D simplified version where the intersection points of the interface lines coincide with grid points in uniform Cartesian grids. Our findings reveal interesting and important results: (1) When the coefficient functions exhibit either high jumps with low-frequency oscillations or low jumps with high-frequency oscillations, the finite element method and finite difference method yield similar numerical solutions. (2) However, when the interface problems involve high-contrast and high-frequency coefficient functions, the numerical solutions obtained from the finite element and finite difference methods differ significantly. Given that the widely studied SPE10 benchmark problem (see https://www.spe.org/web/csp/datasets/set02.htm) typically involves high-contrast and high-frequency permeability due to varying geological layers in porous media, this phenomenon warrants attention. Furthermore, this observation is particularly important for developing multiscale methods, as reference solutions for these methods are usually obtained using the standard second-order finite element method with a fine mesh, and analytical solutions are not available. We provide sufficient details to enable replication of our numerical results, and the implementation is straightforward. This simplicity ensures that readers can easily confirm the validity of our findings.

翻译：本文讨论了用于数值求解椭圆交叉界面问题的二阶有限元法（FEM）与有限差分法（FDM）。该问题具有以下特征：界面由垂直与水平直线构成，系数分段常数，满足两个齐次跳跃条件，源项连续，并采用Dirichlet边界条件。为简洁起见，我们考虑一个二维简化模型，其中界面直线的交点与均匀笛卡尔网格的网格点重合。我们的研究揭示了有趣且重要的结果：（1）当系数函数呈现高跳跃低频振荡或低跳跃高频振荡特性时，有限元法与有限差分法得到的数值解相近。（2）然而，当界面问题涉及高对比度与高频振荡的系数函数时，有限元法与有限差分法所得的数值解存在显著差异。鉴于广泛研究的SPE10基准问题（参见 https://www.spe.org/web/csp/datasets/set02.htm）通常因多孔介质中不同地质层而呈现高对比度与高频渗透率，这一现象值得关注。此外，这一观察对于发展多尺度方法尤为重要，因为这类方法的参考解通常采用标准二阶有限元法在细网格上获得，且解析解不可得。我们提供了充分的细节以确保数值结果的可复现性，且实现过程简单直接。这种简洁性确保了读者能够轻松验证我们结论的有效性。