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.

Keywords: multiscale, mixed finite element, mortar finite element, multipoint flux approximation, cell-centered finite difference, full tensor coefficient, multiblock, nonmatching grids, quadrilaterals, hexahedra

@article{M2AN_2012__46_4_759_0, author = {Wheeler, Mary Fanett and Xue, Guangri and Yotov, Ivan}, title = {A multiscale mortar multipoint flux mixed finite element method}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis }, pages = {759--796}, publisher = {EDP-Sciences}, volume = {46}, number = {4}, year = {2012}, doi = {10.1051/m2an/2011064}, mrnumber = {2891469}, zbl = {1275.65082}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/m2an/2011064/} }

TY - JOUR AU - Wheeler, Mary Fanett AU - Xue, Guangri AU - Yotov, Ivan TI - A multiscale mortar multipoint flux mixed finite element method JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2012 SP - 759 EP - 796 VL - 46 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2011064/ DO - 10.1051/m2an/2011064 LA - en ID - M2AN_2012__46_4_759_0 ER -

%0 Journal Article %A Wheeler, Mary Fanett %A Xue, Guangri %A Yotov, Ivan %T A multiscale mortar multipoint flux mixed finite element method %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2012 %P 759-796 %V 46 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2011064/ %R 10.1051/m2an/2011064 %G en %F M2AN_2012__46_4_759_0

Wheeler, Mary Fanett; Xue, Guangri; Yotov, Ivan. A multiscale mortar multipoint flux mixed finite element method. ESAIM: Mathematical Modelling and Numerical Analysis , Volume 46 (2012) no. 4, pp. 759-796. doi : 10.1051/m2an/2011064. http://archive.numdam.org/articles/10.1051/m2an/2011064/

[1] A hierarchical multiscale method for two-phase flow based on mixed finite elements and nonuniform coarse grids. Multiscale Model. Simul. 5 (2006) 337-363. | MR | Zbl

, and ,[2] Mixed multiscale finite element methods using limited global information. Multiscale Model. Simul. 7 (2008) 655-676. | MR | Zbl

, and ,[3] An introduction to multipoint flux approximations for quadrilateral grids. Comput. Geosci. 6 (2002) 405-432. | MR | Zbl

,[4] Discretization on unstructured grids for inhomogeneous, anisotropic media. I. Derivation of the methods. SIAM J. Sci. Comput. 19 (1998) 1700-1716. | MR | Zbl

, , and ,[5] Convergence of a symmetric MPFA method on quadrilateral grids. Comput. Geosci. 11 (2007) 333-345. | MR | Zbl

, , , and ,[6] A compact multipoint flux approximation method with improved robustness. Numer. Methods for Partial Differential Equations 24 (2008) 1329-1360. | MR | Zbl

, , and ,[7] The G method for heterogeneous anisotropic diffusion on general meshes. Math. Model. Numer. Anal. 44 (2010) 597-625. | Numdam | MR | Zbl

, and ,[8] Analysis of a two-scale, locally conservative subgrid upscaling for elliptic problems. SIAM. J. Numer. Anal. 42 (2004) 576-598. | MR | Zbl

,[9] Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite differences. SIAM J. Numer. Anal. 34 (1997) 828-852. | MR | Zbl

, and ,[10] Enhanced cell-centered finite differences for elliptic equations on general geometry. SIAM J. Sci. Comput. 19 (1998) 404-425. | MR | Zbl

, , , and ,[11] Mixed finite element methods on nonmatching multiblock grids. SIAM J. Numer. Anal. 37 (2000) 1295-1315. | MR | Zbl

, , and ,[12] A multiscale mortar mixed finite element method. Multiscale Model. Simul. 6 (2007) 319-346. | MR

, , and ,[13] Mixed and nonconforming finite element methods : implementation, postprocessing and error estimates. RAIRO Modèl. Math. Anal. Numèr. 19 (1985) 7-32. | Numdam | MR | Zbl

and ,[14] Quadrilateral H(div) finite elements. SIAM J. Numer. Anal. 42 (2005) 2429-2451. | MR | Zbl

, and ,[15] Connection between finite volume and mixed finite element methods. RAIRO Modèl. Math. Anal. Numèr. 30 (1996) 445-465. | Numdam | MR | Zbl

, and ,[16] A new nonconforming approach to domain decomposition : The mortar element method, in Nonlinear Partial Differential Equations and Their Applications, edited by H. Brezis and J.L. Lions. Longman Scientific and Technical, Harlow, UK (1994). | MR | Zbl

, and .[17] Superconvergence of the velocity in mimetic finite difference methods on quadrilaterals. SIAM. J. Numer. Anal. 43 (2005) 1728-1749. | MR | Zbl

, , , and ,[18] The Mathematical Theory of Finite Element Methods. Texts in Applied Mathematics, Springer-Verlag (2007). | Zbl

and ,[19] Mixed and hybrid finite element methods. Springer-Verlag, New York (1991). | MR | Zbl

and ,[20] Two families of mixed finite elements for second order elliptic problems. Numer. Math. 47 (1985) 217-235. | MR | Zbl

, and ,[21] Mixed finite elements for second order elliptic problems in three variables. Numer. Math. 51 (1987) 237-250. | MR | Zbl

, , and ,[22] Error analysis of piecewise constant pressure approximations of Darcy's law. Comput. Methods Appl. Mech. Engrg. 195 (2006) 1547-1559. | MR | Zbl

, and ,[23] Control-volume mixed finite element methods. Comput. Geosci. 1 (1997) 289-315 (1998). | MR | Zbl

, , and ,[24] A mixed multiscale finite element method for elliptic problems with oscillating coefficients. Math. Comp. 72 (2003) 541-576. | MR | Zbl

and ,[25] The Finite Element Method for Elliptic Problems, Stud. Math. Appl. 4. North-Holland, Amsterdam (1978); reprinted, SIAM, Philadelphia (2002). | MR | Zbl

,[26] Superconvergence for rectangular mixed finite elements. Numer. Math. 58 (1990) 287-298. | MR | Zbl

,[27] Unstructured, control-volume distributed, full-tensor finite-volume schemes with flow based grids. Comput. Geosci. 6 (2002) 433-452. | MR | Zbl

,[28] Finite volume discretization with imposed flux continuity for the general tensor pressure equation. Comput. Geosci. 2 (1998) 259-290 (1999). | MR | Zbl

and ,[29] Superconvergence of mixed finite element approximations over quadrilaterals. SIAM. J. Numer. Anal. 36 (1999) 772-787. | MR | Zbl

, and ,[30] Finite volume methods. in Handbook of Numerical Analysis. North-Holland, Amsterdam (2000) 713-1020. | MR | Zbl

, and ,[31] An introduction to the mathematical theory of the Navier-Stokes equations I. Linearized steady problems, Springer-Verlag, New York (1994) | MR | Zbl

,[32] Implementation of a mortar mixed finite element using a multiscale flux basis. Comput. Methods Appl. Mech. Engrg. 198 (2009) 3989-3998. | MR | Zbl

and ,[33] Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms. Springer-Verlag (1986). | MR | Zbl

and ,[34] Domain decomposition and mixed finite element methods for elliptic problems, in First International Symposium on Domain Decomposition Methods for Partial Differential Equations, edited by R. Glowinski, G.H. Golub, G.A. Meurant and J. Periaux. SIAM, Philadelphia (1988) 144-172. | MR | Zbl

and ,[35] Elliptic Problems in Nonsmooth Domains. Pitman, Boston, MA (1995). | Zbl

,[36] A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys. 134 (1997) 169-189. | MR | Zbl

and ,[37] The variational multiscale method-a paradim for computational mechanics. Comput. Methods Appl. Mech. Engrg. 166 (1998) 3-24. | MR | Zbl

, , and ,[38] The numerical solution of diffusion problem in strongly heterogeneous non-isotropic materials. J. Comput. Phys. 132 (1997) 130-148. | MR | Zbl

, and ,[39] A multipoint flux mixed finite element method on hexahedra. SIAM J. Numer. Anal. 48 (2010) 1281-1312. | MR | Zbl

, and ,[40] Multi-scale finite-volume method for elliptic problems in subsurface flow simulation. J. Comput. Phys. 187 (2003) 47-67. | Zbl

, and ,[41] Robust convergence of multi point flux approximation on rough grids. Numer. Math. 104 (2006) 317-337. | MR | Zbl

and ,[42] Convergence of multipoint flux approximations on quadrilateral grids. Numer. Methods Partial Differential Equations 22 (2006) 1438-1454. | MR | Zbl

and ,[43] Non-homogeneous Boundary Value Problems and Applications. Springer-Verlag, Berlin, Heidelberg, New York (1972). | Zbl

and ,[44] Local flux mimetic finite difference methods. Numer. Math. 112 (2009) 115-152. | MR | Zbl

, and ,[45] Domain Decomposition and Iterative Methods for Mixed Finite Element Discretizations of Elliptic Problems. Tech. Report 463, Courant Institute of Mathematical Sciences, New York University, New York (1989). | MR

,[46] Mixed finite elements in R3. Numer. Math. 35 (1980) 315-341. | MR | Zbl

,[47] Balancing domain decomposition for mortar mixed finite element methods on non-matching grids. Numer. Linear Algebra Appl. 10 (2003) 159-180. | MR | Zbl

and ,[48] A mixed finite element method for 2-nd order elliptic problems, in Mathematical aspects of the Finite Elements Method, Lect. Notes Math. 606 (1977) 292-315. | MR | Zbl

and ,[49] Mixed and hybrid methods, in Handbook of Numerical Analysis II, edited by P.G. Ciarlet and J.L. Lions. Elsevier Science Publishers B.V. (1991) 523-639. | MR | Zbl

and ,[50] Finite element and finite difference methods for continuous flows in porous media, in The Mathematics of Reservoir Simulation, edited by R.E. Ewing. SIAM, Philadelphia (1983) 35-106. | Zbl

and ,[51] Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp. 54 (1990) 483-493. | MR | Zbl

and ,[52] These de Doctorat d'etat, Sur l'analyse numérique des méthodes d'éléments finis hybrides et mixtes. Ph.D. thesis, à l'Université Pierre et Marie Curie (1977).

,[53] Equivalence between lowest-order mixed finite element and multi-point finite volume methods on simplicial meshes. ESAIM :M2AN 40 (2006) 367-391. | Numdam | MR | Zbl

,[54] Mixed finite element method over quadrilaterals, in Conference on Advances in Numerical Methods and Applications, edited by I.T. Dimov, B. Sendov and P. Vassilevski. World Scientific, River Edge, NJ (1994) 351-375. | Zbl

and ,[55] On convergence of block-centered finite differences for elliptic problems. SIAM J. Numer. Anal. 25 (1988) 351-375. | MR | Zbl

and ,[56] A multipoint flux mixed finite element method. SIAM. J. Numer. Anal. 44 (2006) 2082-2106. | MR | Zbl

and ,[57] A multipoint flux mixed finite element method on distorted quadrilaterals and hexahedra. Accepted by Numer. Math. (2011). | Zbl

, and ,[58] From mixed finite elements to finite volumes for elliptic PDEs in two and three dimensions. Internat. J. Numer. Methods Engrg. 59 (2004) 365-388. | MR | Zbl

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