Online Adaptive Basis Enrichment for Mixed CEM-GMsFEM

Abstract

In this research, an online basis enrichment strategy for the constraint energy minimizing generalized multiscale finite element method in mixed formulation is proposed. The online approach is based on the technique of oversampling. One makes use of the information of residual and the data in the partial differential equation such as the source function. The analysis presented shows that the proposed online enrichment leads to a fast convergence from multiscale approximation to the fine-scale solution. The error reduction can be made sufficiently large by suitably selecting oversampling regions and the number of oversampling layers. Also, the convergence rate of the enrichment can be tuned by a user-defined parameter. Numerical results are provided to illustrate the efficiency of the proposed method.

A new certified hierarchical and adaptive RB-ML-ROM surrogate model for parametrized PDEs

“…To overcome these shortcomings, adaptive enrichment has been suggested to incrementally refine initially coarse ROMs (cf. [45,12,9,10,65,40]). Thus, we aim at not employing ROMs directly, but rather to leverage (a hierarchy of) ROMs to provide computationally efficient algorithms yielding efficient approximations of prescribed accuracy in an adaptive manner.…”

“…Therefore, adaptive enrichment in model order reduction (MOR) has been suggested to incrementally refine initially coarse ROMs. Adaptive enrichment has first been proposed in the context of localized MOR [45], where successively also concepts to combine offline training and online enrichment were discussed [12,9,10,65,40]. In the context of particular multi-query applications, such as PDE constrained parameter optimization, adaptive enrichment has recently seen great success also for global model reduction approaches, e.g.…”

Multiscale modeling and analysis for some special additive noises driven stochastic partial differential equations

“…The variational multiscale method [12], and residual‐free bubble FEM [13] had used functions related to microstructure characteristics to expand the basis functions of the finite element space. Another way was to construct the finite element space by the multiscale basis function, which includes the Two‐scale FEM [14], multiscale finite element method (MsFEM) [15], generalized multiscale finite element method (GMsFEM) [16], and constraint energy minimizing generalized multiscale finite element method (CEM‐GMsFEM) [17, 18], and so forth. In this paper, our approach is based on the framework of CEM‐GMsFEM, and the constructions of multiscale basis functions are formulated as follows.…”

“…Then, the L 2 -norm of u n − u ms n will be stated and proved for the solutions u n of the elliptic problem (17) and the solutions u ms n ∈ V ms of the multiscale problem (18). We first state a bound of u n in the following lemma.…”

Online adaptive algorithm for Constraint Energy Minimizing Generalized Multiscale Discontinuous Galerkin Method

“…In this research, we develop and analyze an online enrichment strategy for CEM-GMsFEM within the DG setting. The strategy is based on the information of local residuals and the technique of oversampling, adopting the ideas presented in [19] with CG setting and in [24] for mixed formulation. As a result, the corresponding online basis functions are supported in some oversampled regions.…”