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GMsFEM for Nonlinear Problems & Space-Time GMsFEM
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Many engineering and scientific applications deal with models that have multiple spatial scales, and these scales can be non-separable. Many of these processes can exhibit nonlinearities and have a tight coupling with the temporal scales. Because of the scale disparity, modeling these processes in the fine-scale approaches often involves the use of a large number of degrees of freedom, and thus can be prohibitively expensive. As such, some efficient model reduction methods are required to handle the multiscale problems. In the dissertation, we are solving multiscale nonlinear problems and time-dependent problems. Existing methods to solve these problems include numerical homogenization, multiscale finite element methods, heterogeneous multiscale methods, and the Generalized Multiscale Finite Element Methods (GMsFEM). GMsFEM approaches propose a systematic enrichment, which calculates multiscale basis functions via local spectral decomposition in each coarse cell. We first propose a multiscale model reduction framework within GMsFEM for nonlinear elliptic problems. We consider an exemplary problem, which consists of nonlinear p-Laplacian with heterogeneous coefficients. The main challenging feature of this problem is that local subgrid models are nonlinear involving the gradient of the solution. Our novel work includes re-casting the multiscale model reduction problem onto the boundaries of coarse cells, and introducing nonlinear eigenvalue problems in the snapshot space for these nonlinear “harmonic” functions. We also present convergence analysis and numerical results, which show that our approaches can recover the fine-scale solution with a few degrees of freedom. The proposed methods can, in general, be used for more general nonlinear problems, where one needs nonlinear local spectral decomposition. We next consider solving problems with multiple scales in space and time. We develop our approaches within the frameworks of GMsFEM and Generalized Multiscale Discontinuous Galerkin Methods (GMsDGM), separately, using space-time coarse cells. Previous research in developing multiscale spaces within GMsFEM or GMsDGM mainly considered spatial multiscale spaces and relevant ingredients only, which will usually lead to very high dimensional models. In the dissertation, we construct space-time offline spaces, where local spectral decomposition methods are designed based on our analysis. We also discuss adding the online stage, where we make use of the online residual information to construct online basis functions. Numerical results are presented to verify the theoretical findings and to show that using our proposed approaches, we can obtain an accurate solution with low dimensional coarse spaces.
Ye, Shuai (2016). GMsFEM for Nonlinear Problems & Space-Time GMsFEM. Doctoral dissertation, Texas A & M University. Available electronically from