We present a multiscale finite element method for a diffusion problem with rough and high contrast coefficients. The construction of the multiscale finite element space is based on the localized orthogonal decomposition methodology and it involves solutions of local finite element eigenvalue problems. We show that the performance of the multiscale finite element method is similar to the performance of standard finite element methods for the homogeneous Dirichlet boundary value problem for the Poisson equation on smooth or convex domains.} Simple explicit error estimates are established under conditions that can be verified from the outputs of the computation.

翻译：本文针对具有粗糙且高对比度系数的扩散问题，提出了一种多尺度有限元方法。该多尺度有限元空间的构建基于局部正交分解方法，涉及求解局部有限元特征值问题。研究表明，对于光滑或凸区域上的泊松方程齐次狄利克雷边值问题，该多尺度有限元方法的性能与标准有限元方法相当。在可通过计算输出验证的条件下，建立了简洁的显式误差估计。