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:我们简要审查了在不进行比例分离和高频波传播的多元介质中有效减少PDE模型的指数趋同多级有限元素法(ExmmsFEM),在古典MsFEM中,ExmsFEM建在无覆盖的域分解上,同时系统地丰富近似空间,以便在基础功能的数量方面达到近似指数的趋同率。与文献中大多数MsFEM的概括性不同,ExmsFEM并不依赖于任何统一功能的分割。一般而言,必须使用依赖于右侧的功能表示法来打破高频高频科尔莫戈罗夫($n$-width)屏障,以便实现指数趋同。事实上,ExpmSFEM提供的功能表示法中有在线和离线部分。在线部分取决于当地右侧,可以同时进行计算。离线部分包含Galerkin方法中用来集合坚硬度矩阵的基础功能;它们都独立于右侧,因此坚度矩阵可以反复在多式假设中使用。