Generalized Multiscale Finite Element Methods with energy minimizing oversampling
Generalized Multiscale Finite Element Methods with energy minimizing oversampling
复制标题
具有能量最小化过采样的广义多尺度有限元方法
DOI：
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发表时间：
2018
影响因子：
2.9
通讯作者：
W. Leung
中科院分区：
文献类型：
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作者：
Eric T. Chung;Y. Efendiev;W. Leung
本文提出了在广义多尺度有限元法中利用过采样和稳定分解构造多尺度基函数的一般概念。过采样是指使用更大的区域来构造多尺度基函数，稳定分解可以估计局部误差。多尺度方法的分析包括将误差分解为粗区域，在粗区域中估计每个误差贡献。在这种估计中，我们经常使用过采样技术来实现快速收敛。我们在混合离散、内罚间断伽辽金离散和杂交间断伽辽金离散中展示了我们的概念。该基函数的一个重要特点是可用于在线广义多尺度有限元法，即利用残差构造多尺度基函数。在这些问题中，实现快速收敛是很重要的，如果我们有一个稳定的分解，就可以保证快速收敛。在我们的数值结果中，我们给出了离线和在线多尺度基函数的例子。数值结果表明，使用在线基函数可以实现快速收敛。此外，由于使用了多尺度胶合函数，使用杂交间断Galerkin的耦合比使用内罚间断Galerkin的耦合具有更好的精度。
In this paper, we propose a general concept for constructing multiscale basis functions within Generalized Multiscale Finite Element Method, which uses oversampling and stable decomposition. The oversampling refers to using larger regions in constructing multiscale basis functions and stable decomposition allows estimating the local errors. The analysis of multiscale methods involves decomposing the error by coarse regions, where each error contribution is estimated. In this estimate, we often use oversampling techniques to achieve a fast convergence. We demonstrate our concepts in the mixed, the Interior Penalty Discontinuous Galerkin, and Hybridized Discontinuous Galerkin discretizations. One of the important features of the proposed basis functions is that they can be used in online Generalized Multiscale Finite Element Method, where one constructs multiscale basis functions using residuals. In these problems, it is important to achieve a fast convergence, which can be guaranteed if we have a stable decomposition. In our numerical results, we present examples for both offline and online multiscale basis functions. Our numerical results show that one can achieve a fast convergence when using online basis functions. Moreover, we observe that coupling using Hybridized Discontinuous Galerkin provides a better accuracy compared with Interior Penalty Discontinuous Galerkin, which is due to using multiscale glueing functions.