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.
AlgMortar: a fully algebraic multiscale mortar preconditioner
Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo（圣保罗大学数学与计算科学学院）；Piri Technologies, LLC（Piri科技有限公司）；Department of Mathematical Sciences, The University of Texas at Dallas（德克萨斯大学达拉斯分校数学科学系）
Additive average Schwarz with adaptive coarse spaces: scalable algorithms for multiscale problems
Faculty of Mathematics, Univeristy of Warsaw（华沙大学数学系）；Department of Computing, Mathematics and Physics, Western Norway University of Applied Sciences（西挪威应用科学大学计算、数学与物理系）
A residual driven multiscale method for Darcy's flow in perforated domains
Hunan Key Laboratory for Computation and Simulation in Science and Engineering, National Center for Applied Mathematics in Hunan, Xiangtan University（湘潭大学湖南省计算与仿真科学工程重点实验室、湖南应用数学中心）；Eastern Institute for Advanced Study, Eastern Institute of Technology（宁波东部理工学院先进研究院）；Hunan Research Center of the Basic Discipline Fundamental Algorithmic Theory and Novel Computational Methods, Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education, Xiangtan University（湘潭大学教育部智能计算与信息处理重点实验室、湖南省基础学科基本算法理论与新型计算方法研究中心）；Hunan International Scientific and Technological Innovation Cooperation Base of Computational Science, Xiangtan University（湘潭大学计算科学国际科技创新合作基地）
