| In this work we develop multiscale finite element methods for elliptic problems with rapidly varying coefficients. These problems often arise in models of oil flow in porous media that involve many different scales, from the smallest scale of the pores to the largest scale of the entire oil reservoir. Resolving different scales with good accuracy in reasonable period of computational time is a very challenging task. Direct finite element methods are not very practical when it comes to applications to flows in porous media, since it usually takes a very long time to compute solutions to real world problems. Various numerical methods for simulations of flows in porous media, that take into account the multiscale nature of the problem have been developed (such as heterogeneous multiscale methods, methods based on upscaling techniques and multiscale finite element methods). The scope of the present work is limited to multiscale methods based on finite element approach.; In the first part of this thesis we formulate multiscale FEM based on conforming elements for linear problems, prove existence and uniqueness of the solution of the model problem. We also give the convergence analysis for cases when the grid size resolves fine scales of the problem, and more importantly, when it does not. For latter we utilize some of the results of homogenization theory. Our analysis shows, that when the mesh size coincides with the fine scale, the numerical solution suffers from resonance effects. To reduce them we introduce an oversampling technique. We then move on to show several convergence results for a problem with random coefficients.; In the second part we analyze the multiscale FEM based on nonconforming finite elements. Proofs of existence and uniqueness for the linear problem, as well as convergence analysis, are presented for the case of nonconforming elements. Special attention is given to the discussion of convergence results in situations when the scale of variations of the coefficient is comparable with the mesh size.; The last chapter discusses numerical experiments, validating our theoretical results. We describe the numerical method, then run the simulations for cases when exact solution is known so that convergence can be analyzed. Several different forms of the rapidly varying coefficients in the equation are considered. |
