Convergence of the CEM-GMsFEM for compressible flow in highly heterogeneous media

TL;DR

This paper introduces a new multiscale finite element method for efficiently solving complex nonlinear compressible flow problems in highly heterogeneous media, with proven convergence and adaptive enrichment strategies.

Contribution

The paper develops a CEM-GMsFEM approach with exponential decay basis functions, providing convergence analysis independent of heterogeneity and an online enrichment for improved accuracy.

Findings

Convergence depends only on coarse grid size, not heterogeneity.

Numerical experiments confirm theoretical convergence and efficiency.

The method achieves accurate solutions with reduced computational cost.

Abstract

This paper presents and analyses a Constraint Energy Minimization Generalized Multiscale Finite Element Method (CEM-GMsFEM) for solving single-phase non-linear compressible flows in highly heterogeneous media. The construction of CEM-GMsFEM hinges on two crucial steps: First, the auxiliary space is constructed by solving local spectral problems, where the basis functions corresponding to small eigenvalues are captured. Then the basis functions are obtained by solving local energy minimization problems over the oversampling domains using the auxiliary space. The basis functions have exponential decay outside the corresponding local oversampling regions. The convergence of the proposed method is provided, and we show that this convergence only depends on the coarse grid size and is independent of the heterogeneities. An online enrichment guided by \emph{a posteriori} error estimator is developed to enhance computational efficiency. Several numerical experiments on a three-dimensional case to confirm the theoretical findings are presented, illustrating the performance of the method and giving efficient and accurate numerical.