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Cluster-based generalized multiscale finite element method for elliptic PDEs with random coefficients. (English) Zbl 1415.65251

Summary: We propose a generalized multiscale finite element method (GMsFEM) based on clustering algorithm to study the elliptic PDEs with random coefficients in the multi-query setting. Our method consists of offline and online stages. In the offline stage, we construct a small number of reduced basis functions within each coarse grid block, which can then be used to approximate the multiscale finite element basis functions. In addition, we coarsen the corresponding random space through a clustering algorithm. In the online stage, we can obtain the multiscale finite element basis very efficiently on a coarse grid by using the pre-computed multiscale basis. The new GMsFEM can be applied to multiscale SPDE starting with a relatively coarse grid, without requiring the coarsest grid to resolve the smallest-scale of the solution. The new method offers considerable savings in solving multiscale SPDEs. Numerical results are presented to demonstrate the accuracy and efficiency of the proposed method for several multiscale stochastic problems without scale separation.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35J15 Second-order elliptic equations
35R60 PDEs with randomness, stochastic partial differential equations

Software:

AS 136
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

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