An Enriched Multiscale Mortar Space for High Contrast Flow Problems (Communications in Computational Physics, 2018)
Mortar methods are widely used techniques for discretizations of partialdifferential equations and preconditioners for the algebraic systems resulting from thediscretizations. For problems with high contrast and multiple scales, the standard mortarspaces are not robust, and some enrichments are necessary in order to obtain anefficient and robust mortar space. In this paper, we consider a class of flow problemsin high contrast heterogeneous media, and develop a systematic approach to obtainan enriched multiscale mortar space. Our approach is based on the constructions oflocal multiscale basis functions. The multiscale basis functions are constructed fromlocal problems by following the framework of the Generalized Multiscale Finite ElementMethod (GMsFEM). In particular, we first create a local snapshot space. Then weselect the dominated modes within the snapshot space using an appropriate ProperOrthogonal Decomposition (POD) technique. These multiscale basis functions showbetter accuracy than polynomial basis for multiscale problems. Using the proposedmultiscale mortar space, we will construct a multiscale finite element method to solvethe flow problem on a coarse grid and a preconditioning technique for the fine scalediscretization of the flow problem. In particular, we develop a multiscale mortar mixedfinite element method using the mortar space. In addition, we will design a two-leveladditive preconditioner and a two-level hybrid preconditioner based on the proposedmortar space for the iterative method applied to the fine scale discretization of theflow problem. We present several numerical examples to demonstrate the efficiencyand robustness of our proposed mortar space with respect to both the coarse multiscalesolver and the preconditioners.