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Center for Nonlinear Studies |
In the talk we introduce a family of expanded mixed Multiscale Finite Element Methods (MsFEMs) and their hybridizations for second-order elliptic equations. This formulation expands the standard mixed Multiscale Finite Element formulation in the sense that four unknowns (hybrid formulation) are solved simultaneously: pressure, gradient of pressure, velocity and Lagrange multipliers. We use multiscale basis functions for both the velocity and the gradient of the pressure. In the expanded mixed MsFEM framework, we consider both cases of separable-scale and non-separable spatial scales. We specifically analyze the methods in three categories: periodic separable scales, G-convergence separable scales, and continuum scales. When there is no scale separation, using some global information can improve accuracy forthe expanded mixed MsFEMs. We present rigorous convergence analysis for expanded mixed MsFEMs. The analysis includes both conforming and nonconforming expanded mixed MsFEM. Numerical results are presented for various multiscale models and flows in porous media with shale barriers to illustrate the efficiency of the expanded mixed MsFEMs.
Host: Humberto C Godinez Vazquez, Mathematical Modeling and Analysis Theoretical Division, 5-9188