We consider finite element methods of multiscale type to approximate solutions for two-dimensional symmetric elliptic partial differential equations with heterogeneous $L^\infty$ coefficients. The methods are of Galerkin type and follow the Variational Multiscale and Localized Orthogonal Decomposition--LOD approaches in the sense that it decouples spaces into \emph{multiscale} and \emph{fine} subspaces. In a first method, the multiscale basis functions are obtained by mapping coarse basis functions, based on corners used on primal iterative substructuring methods, to functions of global minimal energy. This approach delivers quasi-optimal a priori error energy approximation with respect to the mesh size, but it is not robust with respect to high-contrast coefficients. In a second method, edge modes based on local generalized eigenvalue problems are added to the corner modes. As a result, optimal a priori error energy estimate is achieved which is mesh and contrast independent. The methods converge at optimal rate even if the solution has minimum regularity, belonging only to the Sobolev space $H^1$.

翻译：我们考虑采用多尺度类型的有限元方法来逼近具有异质$L^\infty$系数的二维对称椭圆型偏微分方程的解。该方法属于Galerkin类型，并遵循变分多尺度与局部正交分解——LOD方法的思路，即将空间解耦为\emph{多尺度}子空间和\emph{精细}子空间。在第一种方法中，多尺度基函数是通过将基于原始迭代子结构方法中使用的角点的粗网格基函数，映射到全局最小能量函数而获得的。此方法在网格尺寸方面提供了拟最优的先验能量误差逼近，但对于高对比度系数不具备鲁棒性。在第二种方法中，将基于局部广义特征值问题的边模态添加到角点模态中。其结果是获得了与网格和对比度无关的最优先验能量误差估计。即使解仅属于Sobolev空间$H^1$，具有最低正则性，该方法仍能以最优速率收敛。