TY - JOUR
T1 - Dual-scale Galerkin methods for Darcy flow
AU - Wang, Guoyin
AU - Scovazzi, Guglielmo
AU - Nouveau, Léo
AU - Kees, Christopher E.
AU - Rossi, Simone
AU - Colomés, Oriol
AU - Main, Alex
PY - 2018/2/1
Y1 - 2018/2/1
N2 - The discontinuous Galerkin (DG) method has found widespread application in elliptic problems with rough coefficients, of which the Darcy flow equations are a prototypical example. One of the long-standing issues of DG approximations is the overall computational cost, and many different strategies have been proposed, such as the variational multiscale DG method, the hybridizable DG method, the multiscale DG method, the embedded DG method, and the Enriched Galerkin method. In this work, we propose a mixed dual-scale Galerkin method, in which the degrees-of-freedom of a less computationally expensive coarse-scale approximation are linked to the degrees-of-freedom of a base DG approximation. We show that the proposed approach has always similar or improved accuracy with respect to the base DG method, with a considerable reduction in computational cost. For the specific definition of the coarse-scale space, we consider Raviart–Thomas finite elements for the mass flux and piecewise-linear continuous finite elements for the pressure. We provide a complete analysis of stability and convergence of the proposed method, in addition to a study on its conservation and consistency properties. We also present a battery of numerical tests to verify the results of the analysis, and evaluate a number of possible variations, such as using piecewise-linear continuous finite elements for the coarse-scale mass fluxes.
AB - The discontinuous Galerkin (DG) method has found widespread application in elliptic problems with rough coefficients, of which the Darcy flow equations are a prototypical example. One of the long-standing issues of DG approximations is the overall computational cost, and many different strategies have been proposed, such as the variational multiscale DG method, the hybridizable DG method, the multiscale DG method, the embedded DG method, and the Enriched Galerkin method. In this work, we propose a mixed dual-scale Galerkin method, in which the degrees-of-freedom of a less computationally expensive coarse-scale approximation are linked to the degrees-of-freedom of a base DG approximation. We show that the proposed approach has always similar or improved accuracy with respect to the base DG method, with a considerable reduction in computational cost. For the specific definition of the coarse-scale space, we consider Raviart–Thomas finite elements for the mass flux and piecewise-linear continuous finite elements for the pressure. We provide a complete analysis of stability and convergence of the proposed method, in addition to a study on its conservation and consistency properties. We also present a battery of numerical tests to verify the results of the analysis, and evaluate a number of possible variations, such as using piecewise-linear continuous finite elements for the coarse-scale mass fluxes.
KW - Darcy flow
KW - Discontinuous Galerkin method
KW - Elliptic problems
KW - Hybridization
KW - Variational multiscale method
UR - http://www.scopus.com/inward/record.url?scp=85033409653&partnerID=8YFLogxK
U2 - 10.1016/j.jcp.2017.10.047
DO - 10.1016/j.jcp.2017.10.047
M3 - Article
AN - SCOPUS:85033409653
VL - 354
SP - 111
EP - 134
JO - Journal of Computational Physics
JF - Journal of Computational Physics
SN - 0021-9991
ER -