Reduced-order multiscale modeling of nonlinear p-Laplacian flows in high-contrast media

Abstract

In this paper, we consider a class of nonlinear flow problems that are modeled through a time-dependent p-Laplacian formulation. Using an approach that combines the Generalized Multiscale Finite Element Method (GMsFEM) and Discrete Empirical Interpolation Method (DEIM), we are able to accurately approximate the nonlinear p-Laplacian solutions at a significantly reduced cost. In particular, GMsFEM allows us to iteratively solve the global problem using a reduced-order model in which a flexible number of multiscale basis functions are used to construct a spectrally enriched coarse-grid solution space. The iterative procedure requires a number of nonlinear functional updates that are simultaneously made more cost efficient through implementation of DEIM. The combined GMsFEM-DEIM approach is shown to be a flexible framework for producing accurate reduced-order descriptions of the nonlinear model for a wide range of parameters and varying levels nonlinearity. A number of numerical examples are presented to illustrate the effectiveness of the proposed methodology.