标题： 广义有限元方法中重叠区域的最优谱逼近 显示英文标题

标题： Optimal Spectral Approximation in the Overlaps for Generalized Finite Element Methods

摘要： 在本文中，我们研究一种广义有限元方法，用于求解具有粗糙系数的二阶椭圆偏微分方程。 该方法利用在局部子域边界周围的环形区域上求解特征值问题来计算局部逼近空间。 与在整块子域上求解特征值问题的相应方法相比，问题规模和所得系统矩阵的带宽显著减少，从而实现了更快的谱计算。 我们证明了所提出方法的局部逼近误差的几乎指数级先验衰减结果，这表明该方法的整体逼近误差几乎指数级衰减。 所提出的方法也可以用作预条件器，只需对我们的理论进行轻微调整即可证明预条件迭代的最优收敛性。 数值实验被展示以支持所提出方法的有效性并研究其系数鲁棒性。 显示英文摘要

摘要： In this paper, we study a generalized finite element method for solving second-order elliptic partial differential equations with rough coefficients. The method uses local approximation spaces computed by solving eigenvalue problems on rings around the boundary of local subdomains. Compared to the corresponding method that solves eigenvalue problems on the whole subdomains, the problem size and the bandwidth of the resulting system matrices are substantially reduced, resulting in faster spectral computations. We prove a nearly exponential a priori decay result for the local approximation errors of the proposed method, which implies the nearly exponential decay of the overall approximation error of the method. The proposed method can also be used as a preconditioner, and only a slight adaptation of our theory is necessary to prove the optimal convergence of the preconditioned iteration. Numerical experiments are presented to support the effectiveness of the proposed method and to investigate its coefficient robustness.