Constraint Energy Minimizing Generalized Multiscale Finite Element Method for dual continuum model

TL;DR

This paper introduces a novel multiscale finite element method for dual continuum models in subsurface applications, effectively capturing high contrast channels and fractures with localized basis functions, ensuring convergence based on coarse mesh size.

Contribution

The paper develops a new constraint energy minimizing generalized multiscale finite element method tailored for dual continuum models with high heterogeneity and contrast.

Findings

Basis functions are localized despite high contrast channels.

Convergence depends only on the coarse mesh size.

Numerical tests demonstrate the method's effectiveness.

Abstract

The dual continuum model serves as a powerful tool in the modeling of subsurface applications. It allows a systematic coupling of various components of the solutions. The system is of multiscale nature as it involves high heterogeneous and high contrast coefficients. To numerically compute the solutions, some types of reduced order methods are necessary. We will develop and analyze a novel multiscale method based on the recent advances in multiscale finite element methods. Our method will compute multiple local multiscale basis functions per coarse region. The idea is based on some local spectral problems, which are important to identify high contrast channels, and an energy minimization principle. Using these concepts, we show that the basis functions are localized, even in the presence of high contrast long channels and fractures. In addition, we show that the convergence of the method depends only on the coarse mesh size. Finally, we present several numerical tests to show the performance.