Anisotropy occurs in a wide range of applications. Examples include porous media, composite materials, heat transfer, and other fields in science and engineering. Due to the anisotropy, the physical property could vary significantly only in certain directions. As such, the discrete problem will have a very large condition number for traditional numerical methods. In addition, many anisotropic materials contain multiple scales and their physical properties could vary in orders of magnitude. These large variations bring an additional small-scale parameter into the problem. Thus, a proper treatment of the anisotropy not only helps to design robust iterative methods, but also provides accurate approximations of the problem. Various well-developed techniques have been used to address anisotropic problems, such as multigrid methods, adaptive methods, and domain decomposition techniques. More recently, a large class of accurate reduced-order methods have been introduced and applied to many applications. These include multiscale finite element, multiscale finite volume, and mixed multiscale finite element methods. The primary focus of this dissertation is to study a multiscale finite element method for the approximation of heterogeneous problems involving high-anisotropy, high-contrast, parameter dependency. First, we design robust two-level domain decomposition preconditioners using multiscale coarse spaces. Next, a general formulation of heterogeneous problem is investigated using this multiscale finite element method. Then, a multilevel multiscale finite element method is proposed and analyzed to reduce the computational cost. Last, this multiscale finite element method is extended to a convection-diffusion problem.
