Adaptive Generalized Multiscale Model Reduction Techniques for Problems in Perforated Domains

Abstract

Multiscale modeling of complex physical phenomena in many areas, including hydrogeology, material science, chemistry and biology, consists of solving problems in highly heterogeneous porous media. In many of these applications, differential equations are formulated in perforated domains which can be considered as the region outside of inclusions or connected bodies of various sizes. Due to complicated geometries of these inclusions, solutions to these problems have multiscale features. Taking into account the uncertainties, one needs to solve these problems extensively many times. Model reduction techniques are significant for problems in perforated domains in order to improve the computational efficiency. There are some existing approaches for model reduction in perforated domains including homogenization, heterogeneous multiscale methods and multiscale finite element methods. These techniques typically consider the case when there is a scale separation or the perforation distribution is periodic, and assume that the solution space can be approximated by the solutions of directional cell problems and the effective equations contain a limited number of effective parameters. For more complicated problems where the effective properties may be richer, we are interested in developing systematic local multiscale model reduction techniques to obtain accurate macroscale representations of the underlying fine-scale problem in highly heterogeneous perforated domains. In this dissertation, based on the framework of Generalized Multiscale Finite Element Method, we develop novel methods and algorithms including (1) development of systematic local model reduction techniques for computing multiscale basis in perforated domains, (2) numerical analysis and exhaustive simulation utilizing the proposed basis functions, (3) design of different applicable global coupling frameworks and (4) applications to various problems with challenging engineering backgrounds. Our proposed methods can significantly advance the computational efficiency and accuracy for multiscale problems in perforated media.