Spectral multiscale finite element for nonlinear flows in highly heterogeneous media: A reduced basis approach

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.