| dc.description.abstract | Many applications involve media that contain multiple scales and physical properties that vary in orders of magnitude. One example is a rock sample, which has many micro-scale features. Most multiscale problems are often parameter-dependent, where the parameters represent variations in medium properties, randomness, or spatial heterogeneities. Because of disparity of scales in multiscale problems, solving such problems is prohibitively expensive.
Among the most popular and developed techniques for efficiently solving the global system arising from a finite element approximation of the underlying problem on a very fine mesh are multigrid methods, multilevel methods, and domain decomposition techniques. More recently, a new large class of accurate reduced-order methods has been introduced and used in various applications. These include Galerkin multiscale finite element methods, mixed multiscale finite element methods, multiscale finite volume methods, and mortar multiscale methods, and so on.
In this dissertation, a multiscale finite element method is studied for the computation of heterogeneous problems involving high-contrast, no-scale separation, parameter dependency and nonlinearities. A general formulation of the elliptic heterogeneous problems is discussed, including an oversampling strategy and randomized snapshots generation for a more efficient and accurate computation. Furthermore, a multiscale adaptive algorithm is proposed and analyzed to reduce the computational cost. Then, this multiscale finite element method is extended to the nonlinear high-contrast elliptic problems. Specifically, both continuous and discontinuous Galerkin formulations are considered. In the end, an application to high-contrast heterogeneous Brinkman flow is analyzed. | en |
