We provide a concise review of the exponentially convergent multiscale finite element method (ExpMsFEM) for efficient model reduction of PDEs in heterogeneous media without scale separation and in high-frequency wave propagation. ExpMsFEM is built on the non-overlapped domain decomposition in the classical MsFEM while enriching the approximation space systematically to achieve a nearly exponential convergence rate regarding the number of basis functions. Unlike most generalizations of MsFEM in the literature, ExpMsFEM does not rely on any partition of unity functions. In general, it is necessary to use function representations dependent on the right-hand side to break the algebraic Kolmogorov $n$-width barrier to achieve exponential convergence. Indeed, there are online and offline parts in the function representation provided by ExpMsFEM. The online part depends on the right-hand side locally and can be computed in parallel efficiently. The offline part contains basis functions that are used in the Galerkin method to assemble the stiffness matrix; they are all independent of the right-hand side, so the stiffness matrix can be used repeatedly in multi-query scenarios.

翻译：本文简要回顾了指数收敛多尺度有限元方法（ExpMsFEM），该方法用于高效约化无尺度分离的异质介质中的偏微分方程以及高频波传播的模型。ExpMsFEM建立在经典MsFEM的非重叠区域分解基础上，同时系统性地丰富逼近空间，以相对于基函数数量实现近乎指数收敛速率。与文献中大多数MsFEM的推广不同，ExpMsFEM不依赖于任何单位分割函数。通常，为了突破代数Kolmogorov $n$宽障碍以实现指数收敛，必须使用依赖右端项的函数表示。实际上，ExpMsFEM提供的函数表示包含在线部分和离线部分。在线部分局部依赖于右端项，并可高效并行计算。离线部分包含伽辽金方法中用于组装刚度矩阵的基函数；这些基函数均与右端项无关，因此刚度矩阵可在多重查询场景中重复使用。