Some Applications of the Generalized Multiscale Finite Element Method

Many materials in nature are highly heterogeneous and their properties can vary at different scales. Direct numerical simulations in such multiscale media are prohibitively expensive and some types of model reduction are needed. Typical model reduction techniques include upscaling and multiscale methods. In upscaling methods, one upscales the multiscale media properties so that the problem can be solved on a coarse grid. In multiscale method, one constructs multiscale basis functions that capture media information and solves the problem on the coarse grid. Generalized Multiscale Finite Element Method (GMsFEM) is a recently proposed model reduction technique and has been used for various practical applications. This method has no assumption about the media properties, which can have any type of complicated structure. In GMsFEM, we first create a snapshot space, and then solve a carefully chosen eigenvalue problem to form the offline space. One can also construct online space for the parameter dependent problems. It is shown theoretically and numerically that the GMsFEM is very efficient for the heterogeneous problems involving high-contrast, no-scale separation. In this dissertation, we apply the GMsFEM to perform model reduction for the steady state elasticity equations in highly heterogeneous media though some of our applications are motivated by elastic wave propagation in subsurface. We will consider three kinds of coupling mechanism for different situations. For more practical purposes, we will also study the applications of the GMsFEM for the frequency domain acoustic wave equation and the Reverse Time Migration (RTM) based on the time domain acoustic wave equation