A multiscale finite element method for elliptic problems in composite materials and porous media
Journal of Computational Physics
Convergence of a Nonconforming Multiscale Finite Element Method
SIAM Journal on Numerical Analysis
An energy-minimizing interpolation for robust multigrid methods
SIAM Journal on Scientific Computing
Finite Element Method for Elliptic Problems
Finite Element Method for Elliptic Problems
Accurate multiscale finite element methods for two-phase flow simulations
Journal of Computational Physics
Journal of Computational Physics
Multiscale finite element methods for high-contrast problems using local spectral basis functions
Journal of Computational Physics
Generalized multiscale finite element methods (GMsFEM)
Journal of Computational Physics
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In this paper, we study multiscale finite element methods for Richards' equation, a mathematical model to describe fluid flow in unsaturated and highly heterogeneous porous media. In order to compute solutions of Richard's equation, one can use numerical homogenization or multiscale methods that use two-grid procedures: a fine-grid that resolves the heterogeneities and a coarse grid where computations are done. The idea is that the coarse solution procedure captures the fine-grid variations of the solution. Since the media has complicated variations inside of coarse-grid blocks, a large error can be generated during the computation of coarse-scale solutions. In this paper, we consider the case of highly varying coefficients where variations can occur within coarse regions we develop accurate multiscale methods. In order to obtain accurate coarse-scale numerical solutions for Richards' equation, we design an effective multiscale method that is able to capture the multiscale features of the solution without discarding the small scale details. With a careful choice of the coarse basis functions for multiscale finite element methods, we can significantly reduce errors. We use coarse basis functions construction that combines local spectral problems and a Reduced Basis (RB) approach. This is an extension to the nonlinear case of the method proposed by Efendiev et al. (2012) that combines spectral constructions of coarse spaces with RB procedures to efficiently solve linear parameter dependent flow problems. The construction of coarse spaces begins with an initial choice of multiscale basis functions supported in coarse regions. These basis functions are complemented using a local, parameter dependent, weighted eigenvalue problem. The obtained basis functions can capture the small scale features of the solution within a coarse-grid block and give us an accurate coarse-scale solution. The RB procedures are used to efficiently solve for all possible flow scenarios encountered in every single iteration of a fixed point iterative method. We present representative numerical experiments.