Adaptivity and Online Basis Construction for Generalized Multiscale Finite Element Methods
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Many problems in application involve media with multiple scale, for example, in composite materials, porous media. These problems are usually computationally challenging since fine grid computation is extremely expensive. Therefore, one may need to develop a coarse grid model reduction for this type of problems. In this dissertation, we will consider a multiscale method called generalized multiscale finite element method (GMsFEM). GMsFEM follows the framework of multiscale finite element method. Instead of using one basis function per coarse grid node, GMsFEM uses several basis functions for one coarse grid node. Since the media is highly heterogeneous and may involves high contrast, having more than one basis function per node is important to reduce the error significantly. Due to the varying heterogeneity in the domain, we may require different numbers of basis functions in different regions. Then the question is how to determine the number of basis functions in each region. In this dissertation, we will discuss an adaptive enrichment algorithm for enriching basis functions for the regions with large error. We will consider two different types of basis function for enrichment. One is using the pre-computed offline basis functions. We call this method offline adaptive enrichment. The other method uses online constructed basis functions called online adaptive enrichment. In applications, non-conforming basis functions can give us more flexibility on gridding. The discontinuous Galerkin method also makes the mass matrix block diagonal, which enhances the computation speed in solving time-dependent problem with an explicit scheme. In this dissertation, we will discuss offline and online adaptive methods for the generalized multiscale discontinuous Galerkin method (GMsDGM). We will also discuss using GMsDGM for simulating wave propagation in heterogeneous media.
Leung, Wing Tat (2017). Adaptivity and Online Basis Construction for Generalized Multiscale Finite Element Methods. Doctoral dissertation, Texas A & M University. Available electronically from