Generalized multiscale hybridizable discontinuous Galerkin (GMsHDG) method for flows in nonlinear porous media

Abstract

In this paper, we present a generalized multiscale hybridizable discontinuous Galerkin (GMsHDG) method for nonlinear porous media. We modified the spectral multiscale HDG framework introduced in Efendiev et al. (2015), to solve nonlinear problems. Also, we give projection-based error analysis on the GMsHDG method to numerically solve a nonlinear parabolic problem. The proposed method has two main ingredients: linearization and generating reduced dimensional multiscale spaces. The GMsHDG method yields that the error decreases when the eigenvalue of the local eigenvalue problem for generating a multiscale space increases, as demonstrated in the mathematical error analysis. Through representative numerical experiments, we confirm the reliability of the error estimations and show that the proposed method is practical and efficient.