Generalized Multiscale Finite Element Method. Symmetric Interior Penalty Coupling

TL;DR

This paper develops multiscale finite element methods within the discontinuous Galerkin framework to efficiently simulate flows in highly heterogeneous porous media, introducing three novel coarse space constructions and analyzing their stability and accuracy.

Contribution

It introduces three new multiscale finite element spaces for DG methods tailored to heterogeneous media, with stability analysis and numerical validation.

Findings

All three methods are stable and provide accurate approximations.

The snapshot-based space achieves comparable accuracy with reduced computational cost.

Error estimates confirm the effectiveness of the proposed multiscale approaches.

Abstract

Motivated by applications to numerical simulation of flows in highly heterogeneous porous media, we develop multiscale finite element methods for second order elliptic equations. We discuss a multiscale model reduction technique in the framework of the discontinuous Galerkin finite element method. We propose three different finite element spaces on the coarse mesh. The first space is based on a local eigenvalue problem that uses a weighted $L_2-$norm for computing the "mass" matrix. The second space is generated by amending the eigenvalue problem of the first case with a term related to the penalty. The third choice is based on generation of a large space of snapshots and subsequent selection of a subspace of a reduced dimension. The approximation with these spaces is based on the discontinuous Galerkin finite element method framework. We investigate the stability and derive error estimates for the methods and further experimentally study their performance on a representative number of numerical examples.