In this article we develop the Constraint Energy Minimizing Generalized Multiscale Finite Element Method (CEM-GMsFEM) for elliptic partial differential equations with inhomogeneous Dirichlet, Neumann, and Robin boundary conditions, and the high contrast property emerges from the coefficients of elliptic operators and Robin boundary conditions. By careful construction of multiscale bases of the CEM-GMsFEM, we introduce two operators $\mathcal{D}^m$ and $\mathcal{N}^m$ which are used to handle inhomogeneous Dirichlet and Neumann boundary values and are also proved to converge independently of contrast ratios as enlarging oversampling regions. We provide a priori error estimate and show that oversampling layers are the key factor in controlling numerical errors. A series of experiments are conducted, and those results reflect the reliability of our methods even with high contrast ratios.
翻译:在本文中,我们开发了“限制能源以最小化总规模多功能元素(CEM-GMSFEM) ”, 用于具有不均匀的二分方程式、 Neumann 和 Robin 边界条件的椭圆形部分差分方程式, 以及由椭圆形操作员和Robin边界条件的系数产生的高对比特性。 通过仔细构建 CEM- GMSFEM 的多尺度基点, 我们引入了两个操作员$\mathcal{D ⁇ m} 和$\mathcal{N ⁇ m$, 用于处理不均匀的二分点和 Neumann 边界值, 也证明它们与扩大过度采样区域之间的对比比率是独立的。 我们提供了前期误差估计, 并表明过量的层是控制数字错误的关键因素。 我们进行了一系列实验, 这些实验的结果反映了我们方法的可靠性, 即使对比率很高。