A multiscale mortar multipoint flux mixed finite element method
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 46 (2012) no. 4, p. 759-796

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.

DOI : https://doi.org/10.1051/m2an/2011064
Classification:  65N06,  65N12,  65N15,  65N22,  65N30,  76S05
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 - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {46},
number = {4},
year = {2012},
pages = {759-796},
doi = {10.1051/m2an/2011064},
zbl = {1275.65082},
mrnumber = {2891469},
language = {en},
url = {http://www.numdam.org/item/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 - Modélisation Mathématique et Analyse Numérique, Volume 46 (2012) no. 4, pp. 759-796. doi : 10.1051/m2an/2011064. http://www.numdam.org/item/M2AN_2012__46_4_759_0/

[1] J.E. Aarnes, S. Krogstad and K.-A. Lie, 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 2247754 | Zbl 1124.76022

[2] J.E. Aarnes, Y. Efendiev and L. Jiang, Mixed multiscale finite element methods using limited global information. Multiscale Model. Simul. 7 (2008) 655-676. | MR 2443007 | Zbl 1277.76036

[3] I. Aavatsmark, An introduction to multipoint flux approximations for quadrilateral grids. Comput. Geosci. 6 (2002) 405-432. | MR 1956024 | Zbl 1094.76550

[4] I. Aavatsmark, T. Barkve, Ø. Bøe and T. Mannseth, Discretization on unstructured grids for inhomogeneous, anisotropic media. I. Derivation of the methods. SIAM J. Sci. Comput. 19 (1998) 1700-1716. | MR 1618761 | Zbl 0951.65080

[5] I. Aavastsmark, G.T. Eigestad, R.A. Klausen, M.F. Wheeler and I. Yotov, Convergence of a symmetric MPFA method on quadrilateral grids. Comput. Geosci. 11 (2007) 333-345. | MR 2361457 | Zbl 1128.65093

[6] I. Aavatsmark, G.T. Elgestad, B.T. Mallison and J.M. Nordbotten, A compact multipoint flux approximation method with improved robustness. Numer. Methods for Partial Differential Equations 24 (2008) 1329-1360. | MR 2427194 | Zbl 1230.65114

[7] L. Agélas, D.A. Di Pietro and J. Droniou, The G method for heterogeneous anisotropic diffusion on general meshes. Math. Model. Numer. Anal. 44 (2010) 597-625. | Numdam | MR 2683575 | Zbl 1202.65143

[8] T. Arbogast, Analysis of a two-scale, locally conservative subgrid upscaling for elliptic problems. SIAM. J. Numer. Anal. 42 (2004) 576-598. | MR 2084227 | Zbl 1078.65092

[9] T. Arbogast, M.F. Wheeler and I. Yotov, Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite differences. SIAM J. Numer. Anal. 34 (1997) 828-852. | MR 1442940 | Zbl 0880.65084

[10] T. Arbogast, C.N. Dawson, P.T. Keenan, M.F. Wheeler and I. Yotov, Enhanced cell-centered finite differences for elliptic equations on general geometry. SIAM J. Sci. Comput. 19 (1998) 404-425. | MR 1618879 | Zbl 0947.65114

[11] T. Arbogast, L.C. Cowsar, M.F. Wheeler and I. Yotov, Mixed finite element methods on nonmatching multiblock grids. SIAM J. Numer. Anal. 37 (2000) 1295-1315. | MR 1756426 | Zbl 1001.65126

[12] T. Arbogast, G. Pencheva, M.F. Wheeler and I. Yotov, A multiscale mortar mixed finite element method. Multiscale Model. Simul. 6 (2007) 319-346. | MR 2306414 | Zbl pre05255539

[13] D.N. Arnold and F. Brezzi, Mixed and nonconforming finite element methods : implementation, postprocessing and error estimates. RAIRO Modèl. Math. Anal. Numèr. 19 (1985) 7-32. | Numdam | MR 813687 | Zbl 0567.65078

[14] D.N. Arnold, D. Boffi and R.S. Falk, Quadrilateral H(div) finite elements. SIAM J. Numer. Anal. 42 (2005) 2429-2451. | MR 2139400 | Zbl 1086.65105

[15] J. Baranger, J.F. Maitre and F. Oudin, Connection between finite volume and mixed finite element methods. RAIRO Modèl. Math. Anal. Numèr. 30 (1996) 445-465. | Numdam | MR 1399499 | Zbl 0857.65116

[16] C. Bernardi, Y. Maday and A.T. Patera. 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 1268898 | Zbl 0797.65094

[17] M. Berndt, K. Lipnikov, M. Shashkov, M.F. Wheeler and I. Yotov, Superconvergence of the velocity in mimetic finite difference methods on quadrilaterals. SIAM. J. Numer. Anal. 43 (2005) 1728-1749. | MR 2182147 | Zbl 1096.76030

[18] S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. Texts in Applied Mathematics, Springer-Verlag (2007). | Zbl 0804.65101

[19] F. Brezzi and M. Fortin, Mixed and hybrid finite element methods. Springer-Verlag, New York (1991). | MR 1115205 | Zbl 0788.73002

[20] F. Brezzi, J. Douglas and L.D. Marini, Two families of mixed finite elements for second order elliptic problems. Numer. Math. 47 (1985) 217-235. | MR 799685 | Zbl 0599.65072

[21] F. Brezzi, J. Douglas, R. Duran and M. Fortin, Mixed finite elements for second order elliptic problems in three variables. Numer. Math. 51 (1987) 237-250. | MR 890035 | Zbl 0631.65107

[22] F. Brezzi, M. Fortin and L.D. Marini, Error analysis of piecewise constant pressure approximations of Darcy's law. Comput. Methods Appl. Mech. Engrg. 195 (2006) 1547-1559. | MR 2203980 | Zbl 1116.76051

[23] Z. Cai, J.E. Jones, S.F. Mccormick and T.F. Russell, Control-volume mixed finite element methods. Comput. Geosci. 1 (1997) 289-315 (1998). | MR 1690491 | Zbl 0941.76050

[24] Z. Chen and T.Y. Hou, A mixed multiscale finite element method for elliptic problems with oscillating coefficients. Math. Comp. 72 (2003) 541-576. | MR 1954956 | Zbl 1017.65088

[25] P.G. Ciarlet, The Finite Element Method for Elliptic Problems, Stud. Math. Appl. 4. North-Holland, Amsterdam (1978); reprinted, SIAM, Philadelphia (2002). | MR 520174 | Zbl 0383.65058

[26] R. Duran, Superconvergence for rectangular mixed finite elements. Numer. Math. 58 (1990) 287-298. | MR 1075159 | Zbl 0691.65076

[27] M.G. Edwards, Unstructured, control-volume distributed, full-tensor finite-volume schemes with flow based grids. Comput. Geosci. 6 (2002) 433-452. | MR 1956025 | Zbl 1036.76034

[28] M.G. Edwards and C.F. Rogers, Finite volume discretization with imposed flux continuity for the general tensor pressure equation. Comput. Geosci. 2 (1998) 259-290 (1999). | MR 1686429 | Zbl 0945.76049

[29] R.E. Ewing, M.M. Liu and J. Wang, Superconvergence of mixed finite element approximations over quadrilaterals. SIAM. J. Numer. Anal. 36 (1999) 772-787. | MR 1681041 | Zbl 0926.65107

[30] R. Eymard, T. Gallouet and R. Herbin, Finite volume methods. in Handbook of Numerical Analysis. North-Holland, Amsterdam (2000) 713-1020. | MR 1804748 | Zbl 0981.65095

[31] G.P. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations I. Linearized steady problems, Springer-Verlag, New York (1994) | MR 1284205 | Zbl 0949.35005

[32] B. Ganis and I. Yotov, Implementation of a mortar mixed finite element using a multiscale flux basis. Comput. Methods Appl. Mech. Engrg. 198 (2009) 3989-3998. | MR 2557486 | Zbl 1231.76145

[33] V. Girault and P.A. Raviart, Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms. Springer-Verlag (1986). | MR 851383 | Zbl 0585.65077

[34] R. Glowinski and M.F. Wheeler, 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 972516 | Zbl 0661.65105

[35] P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman, Boston, MA (1995). | Zbl 0695.35060

[36] T.Y. Hou and X.H. Wu, A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys. 134 (1997) 169-189. | MR 1455261 | Zbl 0880.73065

[37] T.J.R. Hughes, G.R. Feijoo, L. Mazzei and J.-B. Quincy, The variational multiscale method-a paradim for computational mechanics. Comput. Methods Appl. Mech. Engrg. 166 (1998) 3-24. | MR 1660141 | Zbl 1017.65525

[38] J. Hyman, M. Shashkov and S. Steinberg, The numerical solution of diffusion problem in strongly heterogeneous non-isotropic materials. J. Comput. Phys. 132 (1997) 130-148. | MR 1440338 | Zbl 0881.65093

[39] R. Ingram, M.F. Wheeler and I. Yotov, A multipoint flux mixed finite element method on hexahedra. SIAM J. Numer. Anal. 48 (2010) 1281-1312. | MR 2684336 | Zbl 1228.65225

[40] P. Jenny, S.H. Lee and H.A. Tchelepi, Multi-scale finite-volume method for elliptic problems in subsurface flow simulation. J. Comput. Phys. 187 (2003) 47-67. | Zbl 1047.76538

[41] R.A. Klausen and R. Winther, Robust convergence of multi point flux approximation on rough grids. Numer. Math. 104 (2006) 317-337. | MR 2244356 | Zbl 1102.76036

[42] R.A. Klausen and R. Winther, Convergence of multipoint flux approximations on quadrilateral grids. Numer. Methods Partial Differential Equations 22 (2006) 1438-1454. | MR 2257642 | Zbl 1106.76043

[43] J.L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications. Springer-Verlag, Berlin, Heidelberg, New York (1972). | Zbl 0223.35039

[44] K. Lipnikov, M. Shashkov and I. Yotov, Local flux mimetic finite difference methods. Numer. Math. 112 (2009) 115-152. | MR 2481532 | Zbl 1165.65063

[45] T.P. Mathew, 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 2638287

[46] J.C. Nedelec, Mixed finite elements in R3. Numer. Math. 35 (1980) 315-341. | MR 592160 | Zbl 0419.65069

[47] G. Pencheva and I. Yotov, Balancing domain decomposition for mortar mixed finite element methods on non-matching grids. Numer. Linear Algebra Appl. 10 (2003) 159-180. | MR 1964290 | Zbl 1071.65169

[48] P.A. Raviart and J. Thomas, 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 483555 | Zbl 0362.65089

[49] J.E. Roberts and J.-M. Thomas, 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 1115239 | Zbl 0875.65090

[50] T.F. Russell and M.F. Wheeler, 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 0572.76089

[51] R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp. 54 (1990) 483-493. | MR 1011446 | Zbl 0696.65007

[52] J.M. Thomas, 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] M. Vohralík, Equivalence between lowest-order mixed finite element and multi-point finite volume methods on simplicial meshes. ESAIM :M2AN 40 (2006) 367-391. | Numdam | MR 2241828 | Zbl 1116.65121

[54] J. Wang and T.P. Mathew, 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 0813.65120

[55] A. Weiser and M.F. Wheeler, On convergence of block-centered finite differences for elliptic problems. SIAM J. Numer. Anal. 25 (1988) 351-375. | MR 933730 | Zbl 0644.65062

[56] M.F. Wheeler and I. Yotov, A multipoint flux mixed finite element method. SIAM. J. Numer. Anal. 44 (2006) 2082-2106. | MR 2263041 | Zbl 1121.76040

[57] M.F. Wheeler, G. Xue and I. Yotov, A multipoint flux mixed finite element method on distorted quadrilaterals and hexahedra. Accepted by Numer. Math. (2011). | Zbl 1277.65100

[58] A. Younès, P. Ackerer and G. Chavent, 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 2029282 | Zbl 1043.65131