A multiscale mortar multipoint flux mixed finite element method
1 Center for Subsurface Modeling, Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, 78712 TX, USA
2 Center for Subsurface Modeling, Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, 78712 TX, USA
3 Department of Mathematics, University of Pittsburgh, Pittsburgh, 15260 PA, USA
Received: 4 August 2010
In this paper, we develop a multiscale mortar multipoint flux mixed finite element method for second order elliptic problems. The equations in the coarse elements (or subdomains) are discretized on a fine grid scale by a multipoint flux mixed finite element method that reduces to cell-centered finite differences on irregular grids. The subdomain grids do not have to match across the interfaces. Continuity of flux between coarse elements is imposed via a mortar finite element space on a coarse grid scale. With an appropriate choice of polynomial degree of the mortar space, we derive optimal order convergence on the fine scale for both the multiscale pressure and velocity, as well as the coarse scale mortar pressure. Some superconvergence results are also derived. The algebraic system is reduced via a non-overlapping domain decomposition to a coarse scale mortar interface problem that is solved using a multiscale flux basis. Numerical experiments are presented to confirm the theory and illustrate the efficiency and flexibility of the method.
Mathematics Subject Classification: 65N06 / 65N12 / 65N15 / 65N22 / 65N30 / 76S05
Key words: Multiscale / mixed finite element / mortar finite element / multipoint flux approximation / cell-centered finite difference / full tensor coefficient / multiblock / nonmatching grids / quadrilaterals / hexahedra
partially supported by the NSF-CDI under contract number DMS 0835745, the DOE grant DE-FGO2-04ER25617, and the Center for Frontiers of Subsurface Energy Security under Contract No. DE-SC0001114.
supported by Award No. KUS-F1-032-04, made by King Abdullah University of Science and Technology (KAUST).
© EDP Sciences, SMAI, 2012