Reduced order modeling techniques for numerical homogenization methods applied to linear and nonlinear multiscale problems

The characteristic of effective properties of physical processes in heterogeneous media is a basic modeling and computational problem for many applications. As standard numerical discretization of such multiscale problems (e.g. with classical finite element method (FEM)) is often computationally prohibitive, there is a need for a novel computational algorithm able to capture the effective behavior of the physical system without resolving the finest scale in the system on the whole computational domain. In this thesis we propose and analyze a new class of numerical methods that combine the so-called finite element heterogeneous multiscale method (FE-HMM) with reduced order modeling techniques for linear and nonlinear multiscale problems. In the first part of the thesis we generalize the FE-HMM to elliptic problems with an arbitrary number of well-separated scales. We provide a rigorous a priori error analysis of this method that generalizes previous work restricted to two-scale problems. In the second part of the thesis, we develop our new reduced order multiscale method that combines the FE-HMM with reduced basis (RB) method. This method, the reduced basis finite element heterogeneous multiscale method (RB-FE-HMM) provides a significant improvement in computational efficiency compared to the FE-HMM, specially for high dimensional problems or high order methods. A priori and a posteriori error analyses are derived for linear elliptic problems, as well as goal oriented adaptivity techniques. The RB-FE-HMM is then generalized to a class of nonlinear elliptic and parabolic problems. A priori error analysis and extensive computational results for nonlinear problems are also provided.