| dc.description.abstract | Many science and engineering problems exhibit scale disparity and high contrast. The small scale features cannot be omitted in the physical models because they can affect the macroscopic behavior of the problems. However, resolving all the scales in these problems can be prohibitively expensive. As a consequence, some types of model reduction techniques are required to design efficient solution algorithms.
For practical purpose, we are interested in mixed finite element problems as they produce solutions with certain conservative properties. Existing multiscale methods for such problems include the mixed multiscale finite element methods. We show that for complicated problems, the mixed multiscale finite element methods may not be able to produce reliable approximations. This motivates the need of enrichment for coarse spaces.
Two enrichment approaches are proposed, one is based on generalized multiscale finite element methods (GMsFEM), while the other is based on spectral element-based algebraic multigrid (ρAMGe). The former one, which is called mixed GMs- FEM, is developed for both Darcy’s flow and linear elasticity. Application of the algorithm in two-phase flow simulations are demonstrated. For linear elasticity, the algorithm is subtly modified due to the symmetry requirement of the stress tensor.
The latter enrichment approach is based on ρAMGe. The algorithm differs from GMsFEM in that both of the velocity and pressure spaces are coarsened. Due the multigrid nature of the algorithm, recursive application is available, which results in an efficient multilevel construction of the coarse spaces.
Stability, convergence analysis, and exhaustive numerical experiments are carried out to validate the proposed enrichment approaches. Our numerical results show that the proposed methods are more efficient than the conventional methods while still being able to produce reliable solution for our targeted applications such as reservoir simulation. Moreover, the robustness of the mixed GMsFEM for linear elasticity with respect to the high contrast heterogeneity in Poisson ratio is evident from our numerical experiments. Lastly, our empirical results show good speedup and approximation by the proposed multilevel coarsening method. | en |
