论文标题
指数收敛的多尺度有限元法
Exponentially Convergent Multiscale Finite Element Method
论文作者
论文摘要
我们对指数收敛的多尺度有限元方法（EXPMSFEM）进行简短的审查，以有效地降低异质介质中的PDE，而无需大小分离和高频波传播。 expmsfem建立在经典MSFEM中的非拼接域分解上，同时系统地富集了近似空间，以达到有关基本函数数量的几乎指数收敛率。与文献中MSFEM的大多数概括不同，ExpMSFEM不依赖于统一函数的任何分区。 通常，有必要使用取决于右侧的功能表示形式来打破代数Kolmogorov $ n $宽 - 宽障碍，以实现指数收敛。确实，ExpMSFEM提供的功能表示形式中有在线和离线零件。在线部分取决于本地右侧，并且可以有效地平行计算。离线部分包含在Galerkin方法中用于组装刚度矩阵的基础函数。它们都独立于右侧，因此刚度矩阵可以在多Query场景中反复使用。
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.
