Communications in Mathematical Sciences

Volume 18 (2020)

Number 3

Constraint energy minimizing generalized multiscale finite element method for dual continuum model

Pages: 663 – 685



Siu Wun Cheung (Department of Mathematics, Texas A&M University, College Station, Tx., U.S.A.)

Eric T. Chung (Department of Mathematics, Chinese University of Hong Kong, Shatin, N.T., Hong Kong)

Yalchin Efendiev (Department of Mathematics & Institute for Scientific Computation (ISC), Texas A&M University, College Station, Tx., U.S.A.)

Wing Tat Leung (Institute of Computational Engineering and Sciences, University of Texas, Austin, Tx., U.S.A.)

Maria Vasilyeva (Institute for Scientific Computation, Texas A&M University, College Station, Tx., U.S.A.; and Department of Computational Technologies, North-Eastern Federal University, Yakutsk, Republic of Sakha (Yakutia), Russia)


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.


fractured porous media, dual continuum model, multiscale method, model reduction

2010 Mathematics Subject Classification

65M12, 65M60

The second-named author’s work is partially supported by Hong Kong RGC General Research Fund (Project 14304217) and CUHK Direct Grant for Research 2017-18.

The third-named author’s work was partially supported by NSF grants 1620318 and 1934904, and by a mega-grant of the Russian Federation Government (N 14.Y26.31.0013).

The fifth-named author’s work was supported by a mega-grant of the Russian Federation Government (N 14.Y26.31.0013).

Received 28 July 2018

Accepted 18 November 2019

Published 30 June 2020