Year: 2018
Communications in Computational Physics, Vol. 23 (2018), Iss. 2 : pp. 476–499
Abstract
Mortar methods are widely used techniques for discretizations of partial differential equations and preconditioners for the algebraic systems resulting from the discretizations. For problems with high contrast and multiple scales, the standard mortar spaces are not robust, and some enrichments are necessary in order to obtain an efficient and robust mortar space. In this paper, we consider a class of flow problems in high contrast heterogeneous media, and develop a systematic approach to obtain an enriched multiscale mortar space. Our approach is based on the constructions of local multiscale basis functions. The multiscale basis functions are constructed from local problems by following the framework of the Generalized Multiscale Finite Element Method (GMsFEM). In particular, we first create a local snapshot space. Then we select the dominated modes within the snapshot space using an appropriate Proper Orthogonal Decomposition (POD) technique. These multiscale basis functions show better accuracy than polynomial basis for multiscale problems. Using the proposed multiscale mortar space, we will construct a multiscale finite element method to solve the flow problem on a coarse grid and a preconditioning technique for the fine scale discretization of the flow problem. In particular, we develop a multiscale mortar mixed finite element method using the mortar space. In addition, we will design a two-level additive preconditioner and a two-level hybrid preconditioner based on the proposed mortar space for the iterative method applied to the fine scale discretization of the flow problem. We present several numerical examples to demonstrate the efficiency and robustness of our proposed mortar space with respect to both the coarse multiscale solver and the preconditioners.
You do not have full access to this article.
Already a Subscriber? Sign in as an individual or via your institution
Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/cicp.OA-2016-0147
Communications in Computational Physics, Vol. 23 (2018), Iss. 2 : pp. 476–499
Published online: 2018-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 24
Keywords: Multiscale domain decomposition finite elements.
-
A Comparison of Mixed Multiscale Finite Element Methods for Multiphase Transport in Highly Heterogeneous Media
Wang, Yiran | Chung, Eric | Fu, Shubin | Huang, ZhaoqinWater Resources Research, Vol. 57 (2021), Iss. 5
https://doi.org/10.1029/2020WR028877 [Citations: 5] -
Online conservative generalized multiscale finite element method for highly heterogeneous flow models
Wang, Yiran | Chung, Eric | Fu, Shubin | Presho, MichaelComputational Geosciences, Vol. 25 (2021), Iss. 5 P.1837
https://doi.org/10.1007/s10596-021-10074-x [Citations: 2] -
Iterative oversampling technique for constraint energy minimizing generalized multiscale finite element method in the mixed formulation
Cheung, Siu Wun | Chung, Eric | Efendiev, Yalchin | Leung, Wing Tat | Pun, Sai-MangApplied Mathematics and Computation, Vol. 415 (2022), Iss. P.126622
https://doi.org/10.1016/j.amc.2021.126622 [Citations: 0] -
Triple mesh methods and their application to two-phase flow in porous media
Adeyemi, Adedimeji A. | Awotunde, Abeeb A. | Patil, Shirish | Mahmoud, Mohamed N. | Sultan, Abdullah S. | Mohanty, KishoreJournal of Petroleum Science and Engineering, Vol. 212 (2022), Iss. P.110252
https://doi.org/10.1016/j.petrol.2022.110252 [Citations: 0] -
An Adaptive Generalized Multiscale Finite Element Method Based Two-Grid Preconditioner for Large Scale High-Contrast Linear Elasticity Problems
Yang, Yanfang | Fu, Shubin | Chung, Eric T.Journal of Scientific Computing, Vol. 92 (2022), Iss. 1
https://doi.org/10.1007/s10915-022-01869-w [Citations: 0] -
A Hybrid High-Order Method for Highly Oscillatory Elliptic Problems
Cicuttin, Matteo | Ern, Alexandre | Lemaire, SimonComputational Methods in Applied Mathematics, Vol. 19 (2019), Iss. 4 P.723
https://doi.org/10.1515/cmam-2018-0013 [Citations: 15] -
Online Mixed Multiscale Finite Element Method with Oversampling and Its Applications
Yang, Yanfang | Fu, Shubin | Chung, Eric T.Journal of Scientific Computing, Vol. 82 (2020), Iss. 2
https://doi.org/10.1007/s10915-019-01121-y [Citations: 6] -
A local-global multiscale mortar mixed finite element method for multiphase transport in heterogeneous media
Fu, Shubin | Chung, Eric T.Journal of Computational Physics, Vol. 399 (2019), Iss. P.108906
https://doi.org/10.1016/j.jcp.2019.108906 [Citations: 8] -
A locally conservative multiscale method for stochastic highly heterogeneous flow
Wang, Yiran | Chung, Eric | Fu, ShubinComputer Methods in Applied Mechanics and Engineering, Vol. 410 (2023), Iss. P.116020
https://doi.org/10.1016/j.cma.2023.116020 [Citations: 0] -
Extended Dual Mesh Method with applications in two-phase flow in heterogeneous porous media
Adeyemi, Adedimeji A. | Awotunde, Abeeb A. | Liao, Qinzhuo | Mohanty, Kishore | Patil, ShirishJournal of Petroleum Science and Engineering, Vol. 204 (2021), Iss. P.108729
https://doi.org/10.1016/j.petrol.2021.108729 [Citations: 3] -
Multiscale Hybridizable Discontinuous Galerkin Method for Flow Simulations in Highly Heterogeneous Media
Yang, Yanfang | Shi, Ke | Fu, ShubinJournal of Scientific Computing, Vol. 81 (2019), Iss. 3 P.1712
https://doi.org/10.1007/s10915-019-01058-2 [Citations: 4] -
A two-grid preconditioner with an adaptive coarse space for flow simulations in highly heterogeneous media
Yang, Yanfang | Fu, Shubin | Chung, Eric T.Journal of Computational Physics, Vol. 391 (2019), Iss. P.1
https://doi.org/10.1016/j.jcp.2019.03.038 [Citations: 5] -
Robust Linear Domain Decomposition Schemes for Reduced Nonlinear Fracture Flow Models
Ahmed, Elyes | Fumagalli, Alessio | Budiša, Ana | Keilegavlen, Eirik | Nordbotten, Jan M. | Radu, Florin A.SIAM Journal on Numerical Analysis, Vol. 59 (2021), Iss. 1 P.583
https://doi.org/10.1137/19M1268392 [Citations: 3]