In this paper, the generalized finite element method (GFEM) for solving second order elliptic equations with rough coefficients is studied. New optimal local approximation spaces for GFEMs based on local eigenvalue problems involving a partition of unity are presented. These new spaces have advantages over those proposed in [I. Babuska and R. Lipton, Multiscale Model.\;\,Simul., 9 (2011), pp.~373--406]. First, in addition to a nearly exponential decay rate of the local approximation errors with respect to the dimensions of the local spaces, the rate of convergence with respect to the size of the oversampling region is also established. Second, the theoretical results hold for problems with mixed boundary conditions defined on general Lipschitz domains. Finally, an efficient and easy-to-implement technique for generating the discrete $A$-harmonic spaces is proposed which relies on solving an eigenvalue problem associated with the Dirichlet-to-Neumann operator, leading to a substantial reduction in computational cost. Numerical experiments are presented to support the theoretical analysis and to confirm the effectiveness of the new method.

翻译：本文研究了用于解决具有粗系数的第二顺序椭圆方程式的通用有限要素法(GFEM),并介绍了基于涉及团结分割的局部双元值问题的GFEMS新的当地最佳近似空间,这些新空间优于[I. Babuska和R. Lipton, Multiscale Model.\;\\;\,Simul., 9(2011), pp.~373-406]中提议的空间。首先,除了当地近似差在地方空间尺寸方面几乎指数性衰减率差外,还确定了过度采样区域面积的趋同率。第二,理论结果涉及Lipschitz通用域界定的混合边界条件的问题。最后,提出了产生离散的美元-和谐空间的高效和易于实施技术,这有赖于解决与Drichlet-Neumann操作员有关的一个电子价值问题,导致计算成本大幅下降。