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A multiscale mortar multipoint flux mixed finite element method

Published online by Cambridge University Press:  03 February 2012

Mary Fanett Wheeler ,
Guangri Xue  and
Ivan Yotov
Mary Fanett Wheeler
Affiliation:
Center for Subsurface Modeling, Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, 78712 TX, USA. [email protected]
Guangri Xue
Affiliation:
Center for Subsurface Modeling, Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, 78712 TX, USA; [email protected]
Ivan Yotov
Affiliation:
Department of Mathematics, University of Pittsburgh, Pittsburgh, 15260 PA, USA; [email protected]
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Abstract

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

Multiscalemixed finite elementmortar finite elementmultipoint flux approximationcell-centered finite differencefull tensor coefficientmultiblocknonmatching gridsquadrilateralshexahedra
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 46 , Issue 4 , July 2012 , pp. 759 - 796
Copyright
© EDP Sciences, SMAI, 2012

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References

Références

Aarnes, J.E., Krogstad, S. and Lie, K.-A., A hierarchical multiscale method for two-phase flow based on mixed finite elements and nonuniform coarse grids. Multiscale Model. Simul. 5 (2006) 337–363. Google Scholar
Aarnes, J.E., Efendiev, Y. and Jiang, L., Mixed multiscale finite element methods using limited global information. Multiscale Model. Simul. 7 (2008) 655–676. Google Scholar
Aavatsmark, I., An introduction to multipoint flux approximations for quadrilateral grids. Comput. Geosci. 6 (2002) 405–432. Google Scholar
Aavatsmark, I., Barkve, T., Bøe, Ø. and Mannseth, T., Discretization on unstructured grids for inhomogeneous, anisotropic media. I. Derivation of the methods. SIAM J. Sci. Comput. 19 (1998) 1700–1716. Google Scholar
Aavastsmark, I., Eigestad, G.T., Klausen, R.A., Wheeler, M.F. and Yotov, I., Convergence of a symmetric MPFA method on quadrilateral grids. Comput. Geosci. 11 (2007) 333–345. Google Scholar
Aavatsmark, I., Elgestad, G.T., Mallison, B.T. and Nordbotten, J.M., A compact multipoint flux approximation method with improved robustness. Numer. Methods for Partial Differential Equations 24 (2008) 1329–1360. Google Scholar
Agélas, L., Di Pietro, D.A. and Droniou, J., The G method for heterogeneous anisotropic diffusion on general meshes. Math. Model. Numer. Anal. 44 (2010) 597–625. Google Scholar
Arbogast, T., Analysis of a two-scale, locally conservative subgrid upscaling for elliptic problems. SIAM. J. Numer. Anal. 42 (2004) 576–598. Google Scholar
Arbogast, T., Wheeler, M.F. and Yotov, I., Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite differences. SIAM J. Numer. Anal. 34 (1997) 828–852. Google Scholar
Arbogast, T., Dawson, C.N., Keenan, P.T., Wheeler, M.F. and Yotov, I., Enhanced cell-centered finite differences for elliptic equations on general geometry. SIAM J. Sci. Comput. 19 (1998) 404–425. Google Scholar
Arbogast, T., Cowsar, L.C., Wheeler, M.F. and Yotov, I., Mixed finite element methods on nonmatching multiblock grids. SIAM J. Numer. Anal. 37 (2000) 1295–1315. Google Scholar
Arbogast, T., Pencheva, G., Wheeler, M.F. and Yotov, I., A multiscale mortar mixed finite element method. Multiscale Model. Simul. 6 (2007) 319–346. Google Scholar
Arnold, D.N. and Brezzi, F., Mixed and nonconforming finite element methods : implementation, postprocessing and error estimates. RAIRO Modèl. Math. Anal. Numèr. 19 (1985) 7–32. Google Scholar
Arnold, D.N., Boffi, D. and Falk, R.S., Quadrilateral H(div) finite elements. SIAM J. Numer. Anal. 42 (2005) 2429–2451. Google Scholar
Baranger, J., Maitre, J.F. and Oudin, F., Connection between finite volume and mixed finite element methods. RAIRO Modèl. Math. Anal. Numèr. 30 (1996) 445–465. Google Scholar
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).
Berndt, M., Lipnikov, K., Shashkov, M., Wheeler, M.F. and Yotov, I., Superconvergence of the velocity in mimetic finite difference methods on quadrilaterals. SIAM. J. Numer. Anal. 43 (2005) 1728–1749. Google Scholar
S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. Texts in Applied Mathematics, Springer-Verlag (2007).
F. Brezzi and M. Fortin, Mixed and hybrid finite element methods. Springer-Verlag, New York (1991).
Brezzi, F., Douglas, J. and Marini, L.D., Two families of mixed finite elements for second order elliptic problems. Numer. Math. 47 (1985) 217–235. Google Scholar
Brezzi, F., Douglas, J., Duran, R. and Fortin, M., Mixed finite elements for second order elliptic problems in three variables. Numer. Math. 51 (1987) 237–250. Google Scholar
Brezzi, F., Fortin, M. and Marini, L.D., Error analysis of piecewise constant pressure approximations of Darcy’s law. Comput. Methods Appl. Mech. Engrg. 195 (2006) 1547–1559. Google Scholar
Cai, Z., Jones, J.E., McCormick, S.F. and Russell, T.F., Control-volume mixed finite element methods. Comput. Geosci. 1 (1997) 289–315 (1998). Google Scholar
Chen, Z. and Hou, T.Y., A mixed multiscale finite element method for elliptic problems with oscillating coefficients. Math. Comp. 72 (2003) 541–576. Google Scholar
P.G. Ciarlet, The Finite Element Method for Elliptic Problems, Stud. Math. Appl. 4. North-Holland, Amsterdam (1978); reprinted, SIAM, Philadelphia (2002).
Duran, R., Superconvergence for rectangular mixed finite elements. Numer. Math. 58 (1990) 287–298. Google Scholar
Edwards, M.G., Unstructured, control-volume distributed, full-tensor finite-volume schemes with flow based grids. Comput. Geosci. 6 (2002) 433–452. Google Scholar
Edwards, M.G. and Rogers, C.F., Finite volume discretization with imposed flux continuity for the general tensor pressure equation. Comput. Geosci. 2 (1998) 259–290 (1999). Google Scholar
Ewing, R.E., Liu, M.M. and Wang, J., Superconvergence of mixed finite element approximations over quadrilaterals. SIAM. J. Numer. Anal. 36 (1999) 772–787. Google Scholar
R. Eymard, T. Gallouet and R. Herbin, Finite volume methods. in Handbook of Numerical Analysis. North-Holland, Amsterdam (2000) 713–1020.
G.P. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations I. Linearized steady problems, Springer-Verlag, New York (1994)
Ganis, B. and Yotov, I., Implementation of a mortar mixed finite element using a multiscale flux basis. Comput. Methods Appl. Mech. Engrg. 198 (2009) 3989–3998. Google Scholar
V. Girault and P.A. Raviart, Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms. Springer-Verlag (1986).
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.
P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman, Boston, MA (1995).
Hou, T.Y. and Wu, X.H., A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys. 134 (1997) 169–189. Google Scholar
Hughes, T.J.R., Feijoo, G.R., Mazzei, L. and Quincy, J.-B., The variational multiscale method–a paradim for computational mechanics. Comput. Methods Appl. Mech. Engrg. 166 (1998) 3–24. Google Scholar
Hyman, J., Shashkov, M. and Steinberg, S., The numerical solution of diffusion problem in strongly heterogeneous non-isotropic materials. J. Comput. Phys. 132 (1997) 130–148. Google Scholar
Ingram, R., Wheeler, M.F. and Yotov, I., A multipoint flux mixed finite element method on hexahedra. SIAM J. Numer. Anal. 48 (2010) 1281–1312. Google Scholar
Jenny, P., Lee, S.H. and Tchelepi, H.A., Multi-scale finite-volume method for elliptic problems in subsurface flow simulation. J. Comput. Phys. 187 (2003) 47–67. Google Scholar
Klausen, R.A. and Winther, R., Robust convergence of multi point flux approximation on rough grids. Numer. Math. 104 (2006) 317–337. Google Scholar
Klausen, R.A. and Winther, R., Convergence of multipoint flux approximations on quadrilateral grids. Numer. Methods Partial Differential Equations 22 (2006) 1438–1454. Google Scholar
J.L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications. Springer-Verlag, Berlin, Heidelberg, New York (1972).
Lipnikov, K., Shashkov, M. and Yotov, I., Local flux mimetic finite difference methods. Numer. Math. 112 (2009) 115–152. Google Scholar
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).
Nedelec, J.C., Mixed finite elements in R 3. Numer. Math. 35 (1980) 315–341. Google Scholar
Pencheva, G. and Yotov, I., Balancing domain decomposition for mortar mixed finite element methods on non-matching grids. Numer. Linear Algebra Appl. 10 (2003) 159–180. Google Scholar
Raviart, P.A. and Thomas, J., 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. Google Scholar
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.
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.
Scott, R. and Zhang, S., Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp. 54 (1990) 483–493. Google Scholar
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).
Vohralík, M., Equivalence between lowest-order mixed finite element and multi-point finite volume methods on simplicial meshes. ESAIM :M2AN 40 (2006) 367–391. Google Scholar
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.
Weiser, A. and Wheeler, M.F., On convergence of block-centered finite differences for elliptic problems. SIAM J. Numer. Anal. 25 (1988) 351–375. Google Scholar
Wheeler, M.F. and Yotov, I., A multipoint flux mixed finite element method. SIAM. J. Numer. Anal. 44 (2006) 2082–2106. Google Scholar
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).
Younès, A., Ackerer, P. and Chavent, G., From mixed finite elements to finite volumes for elliptic PDEs in two and three dimensions. Internat. J. Numer. Methods Engrg. 59 (2004) 365–388. Google Scholar