A multiscale mortar multipoint flux mixed finite element method

Mary Fanett Wheeler; Guangri Xue; Ivan Yotov

ESAIM: Mathematical Modelling and Numerical Analysis (2012)

  • Volume: 46, Issue: 4, page 759-796
  • ISSN: 0764-583X

Abstract

top
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.

How to cite

top

Wheeler, Mary Fanett, Xue, Guangri, and Yotov, Ivan. "A multiscale mortar multipoint flux mixed finite element method." ESAIM: Mathematical Modelling and Numerical Analysis 46.4 (2012): 759-796. <http://eudml.org/doc/277847>.

@article{Wheeler2012,
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.},
author = {Wheeler, Mary Fanett, Xue, Guangri, Yotov, Ivan},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Multiscale; mixed finite element; mortar finite element; multipoint flux approximation; cell-centered finite difference; full tensor coefficient; multiblock; nonmatching grids; quadrilaterals; hexahedra; mixed finite element method; second-order elliptic problems; convergence; superconvergence; domain decomposition; numerical experiments},
language = {eng},
month = {2},
number = {4},
pages = {759-796},
publisher = {EDP Sciences},
title = {A multiscale mortar multipoint flux mixed finite element method},
url = {http://eudml.org/doc/277847},
volume = {46},
year = {2012},
}

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
DA - 2012/2//
PB - EDP Sciences
VL - 46
IS - 4
SP - 759
EP - 796
AB - 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.
LA - eng
KW - Multiscale; mixed finite element; mortar finite element; multipoint flux approximation; cell-centered finite difference; full tensor coefficient; multiblock; nonmatching grids; quadrilaterals; hexahedra; mixed finite element method; second-order elliptic problems; convergence; superconvergence; domain decomposition; numerical experiments
UR - http://eudml.org/doc/277847
ER -

References

top
  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.  
  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.  
  3. I. Aavatsmark, An introduction to multipoint flux approximations for quadrilateral grids. Comput. Geosci.6 (2002) 405–432.  
  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.  
  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.  
  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 Equations24 (2008) 1329–1360.  
  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.  
  8. T. Arbogast, Analysis of a two-scale, locally conservative subgrid upscaling for elliptic problems. SIAM. J. Numer. Anal.42 (2004) 576–598.  
  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.  
  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.  
  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.  
  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.  
  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.  
  14. D.N. Arnold, D. Boffi and R.S. Falk, Quadrilateral H(div) finite elements. SIAM J. Numer. Anal.42 (2005) 2429–2451.  
  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.  
  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).  
  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.  
  18. S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. Texts in Applied Mathematics, Springer-Verlag (2007).  
  19. F. Brezzi and M. Fortin, Mixed and hybrid finite element methods. Springer-Verlag, New York (1991).  
  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.  
  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.  
  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.  
  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).  
  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.  
  25. P.G. Ciarlet, The Finite Element Method for Elliptic Problems, Stud. Math. Appl.4. North-Holland, Amsterdam (1978); reprinted, SIAM, Philadelphia (2002).  
  26. R. Duran, Superconvergence for rectangular mixed finite elements. Numer. Math.58 (1990) 287–298.  
  27. M.G. Edwards, Unstructured, control-volume distributed, full-tensor finite-volume schemes with flow based grids. Comput. Geosci.6 (2002) 433–452.  
  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).  
  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.  
  30. R. Eymard, T. Gallouet and R. Herbin, Finite volume methods. in Handbook of Numerical Analysis. North-Holland, Amsterdam (2000) 713–1020.  
  31. G.P. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations I. Linearized steady problems, Springer-Verlag, New York (1994)  
  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.  
  33. V. Girault and P.A. Raviart, Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms. Springer-Verlag (1986).  
  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.  
  35. P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman, Boston, MA (1995).  
  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.  
  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.  
  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.  
  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.  
  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.  
  41. R.A. Klausen and R. Winther, Robust convergence of multi point flux approximation on rough grids. Numer. Math.104 (2006) 317–337.  
  42. R.A. Klausen and R. Winther, Convergence of multipoint flux approximations on quadrilateral grids. Numer. Methods Partial Differential Equations22 (2006) 1438–1454.  
  43. J.L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications. Springer-Verlag, Berlin, Heidelberg, New York (1972).  
  44. K. Lipnikov, M. Shashkov and I. Yotov, Local flux mimetic finite difference methods. Numer. Math.112 (2009) 115–152.  
  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).  
  46. J.C. Nedelec, Mixed finite elements in R3. Numer. Math.35 (1980) 315–341.  
  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.  
  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.  
  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.  
  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.  
  51. R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp.54 (1990) 483–493.  
  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 :M2AN40 (2006) 367–391.  
  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.  
  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.  
  56. M.F. Wheeler and I. Yotov, A multipoint flux mixed finite element method. SIAM. J. Numer. Anal.44 (2006) 2082–2106.  
  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).  
  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.  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.