, Volume 52, Issue 4, pp 543–557 | Cite as

Abstract multiscale-hybrid-mixed methods

  • Alexandre L. MadureiraEmail author


In an abstract setting, we investigate the basic ideas behind the Multiscale Hybrid Mixed (MHM) method, a Domain Decomposition scheme designed to solve multiscale partial differential equations (PDEs) in parallel. As originally proposed, the MHM method starting point is a primal hybrid formulation, which is then manipulated to result in an efficient method that is based on local independent PDEs and a global problem that is posed on the skeleton of the finite element mesh. Recasting the MHM method in a more general framework, we investigate some conditions that yield a well-posed method. We apply the general ideas to different formulations, and, in particular, come up with an interesting and fruitful connection between the Multiscale Finite Element Method and a dual hybrid method. Finally, we propose a method that combines the main ideas of the Discontinuous Enrichment Method and the MHM method.


MHM Mixed method Hybrid method Domain decomposition Finite element Multiscale 

Mathematics Subject Classification

65N12 65N30 


  1. 1.
    Araya, R., Harder, C., Paredes, D., Valentin, F.: Multiscale hybrid-mixed method. SIAM J. Numer. Anal. 51(6), 3505–3531 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Arbogast, T., Pencheva, G., Wheeler, M.F., Yotov, I.: A multiscale mortar mixed finite element method. Multiscale Model. Simul. 6(1), 319–346 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Arbogast, T., Xiao, H.: A multiscale mortar mixed space based on homogenization for heterogeneous elliptic problems. SIAM J. Numer. Anal. 51(1), 377–399 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Arnold, D.N., Brezzi, F.: Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates. RAIROModél.Math. Anal. Numér. 19(1), 7–32 (1985). English, with French summaryzbMATHMathSciNetGoogle Scholar
  5. 5.
    Boffi, D., Brezzi, F., Fortin, M.: Mixed finite element methods and applications, Springer Series in Computational Mathematics, vol. 44. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  6. 6.
    Brezzi, F., Fortin, M.: Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, vol. 15. Springer, New York (1991)CrossRefGoogle Scholar
  7. 7.
    Brezzi, F., Marini, L.D.: Augmented spaces, two-level methods, and stabilizing subgrids, Internat. J. Numer. Methods Fluids 40(1–2), 31–46 (2002). ICFD conference on numerical methods for fluid dynamics (Oxford, 2001)Google Scholar
  8. 8.
    Brezzi, F., Franca, L.P., Russo, A.: Further considerations on residual-free bubbles for advective-diffusive equations. Comput. Methods Appl. Mech. Eng. 166(1–2), 25–33 (1998)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Chu, C.-C., Graham, I.G., Hou, T.-Y.: A new multiscale finite element method for high-contrast elliptic interface problems. Math. Comp. 79(272), 1915–1955 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Ciarlet, P.G.: The finite element method for elliptic problems. Studies in mathematics and its applications, vol. 4. North-Holland Publishing Co., Amsterdam–New York–Oxford (1978)Google Scholar
  11. 11.
    Cockburn, B., Gopalakrishnan, J.: Error analysis of variable degree mixed methods for elliptic problems via hybridization. Math. Comput. 74(252), 1653–1677 (2005). electroniczbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Cockburn, B., Gopalakrishnan, J., Lazarov, R.: Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems. SIAM J. Numer. Anal. 47(2), 1319–1365 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Codina, R., Badia, S.: On the design of discontinuous Galerkin methods for elliptic problems based on hybrid formulations. Comput. Methods Appl. Mech. Eng. 263, 158–168 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Efendiev, Y.R., Hou, T.Y.: Multiscale finite element methods, surveys and tutorials in the applied mathematical sciences. Theory and applications, vol. 4. Springer, New York (2009)Google Scholar
  15. 15.
    Efendiev, Y.R., Lazarov, R., Shi, K.: A multiscale HDG method for second order elliptic equations. Part I. Polynomial and homogenization-based multiscale spaces (2013) arXiv:1310.2827
  16. 16.
    Ern, A., Gurmond, J.-L.: Theory and practice of finite elements. Applied mathematical sciences, vol. 159. Springer, New York (2004)CrossRefGoogle Scholar
  17. 17.
    Farhat, C., Harari, I., Franca, L.P.: The discontinuous enrichment method. Comput. Methods Appl. Mech. Eng. 190(48), 6455–6479 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Franca, L.P., Russo, A.: Deriving upwinding, mass lumping and selective reduced integration by residual-free bubbles. Appl. Math. Lett. 9(5), 83–88 (1996)zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Francisco, A., Ginting, V., Pereira, F., Rigelo, J.: Design and implementation of a multiscale mixed method based on a nonoverlapping domain decomposition procedure. Math. Comput. Simul. 99, 125–138 (2013)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Glowinski, R., Wheeler, M.F.: Domain decomposition and mixed finite element methods for elliptic problems. Partial differential equations (Paris, 1987) SIAM, Philadelphia, pp. 144–172 (1988)Google Scholar
  21. 21.
    Hansbo, P., Lovadina, C., Perugia, I., Sangalli, G.: A Lagrange multiplier method for the finite element solution of elliptic interface problems using non-matching meshes. Numer. Math. 100(1), 91–115 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Harder, C., Madureira, A., Valentin, F.: A hybrid-mixed method for elasticity in two and three-dimensions (2013) (submitted)Google Scholar
  23. 23.
    Harder, C., Paredes, D., Valentin, F.: A family of multiscale hybrid-mixed finite element methods for the Darcy equation with rough coficients. J. Comput. Phys. 245, 107–130 (2013)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Harder, C., Paredes, D., Valentin, F.: On a multiscale hybrid-mixed method for advective-reactive dominated problems with heterogenous coefficients (2013) (submitted)Google Scholar
  25. 25.
    Hou, T.Y.: Numerical approximations to multiscale solutions in partial differential equations, Frontiers in numerical analysis (Durham, 2002) Universitext, Springer, Berlin, pp. 241–301 (2003)Google Scholar
  26. 26.
    Hsiao, G.C., Schnack, E., Wendland, W.L.: Hybrid coupled finite-boundary element methods for elliptic systems of second order. Comput. Methods Appl. Mech. Eng. 190(5–7), 431–485 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    Qi-ya, H., Liang, G.-P., Liu, J.-Z.: Construction of a preconditioner for domain decomposition methods with polynomial Lagrangian multipliers. J. Comput. Math. 19(2), 213–224 (2001)MathSciNetGoogle Scholar
  28. 28.
    Madureira, A., Valentin, F.: A Locking free primal hybrid method for the Reissner-Mindlin plate model (2013) (in preparation)Google Scholar
  29. 29.
    Quarteroni, A., Valli, A.: Numerical approximation of partial differential equations, Springer Series in Computational Mathematics, vol. 23. Springer, Berlin (1994)Google Scholar
  30. 30.
    Roberts, J.E., Thomas, J.-M.: Mixed and hybrid methods. Handbook of numerical analysis, vol. II, pp. 523–639. North-Holland, Amsterdam (1991)Google Scholar
  31. 31.
    Sangalli, G.: Capturing small scales in elliptic problems using a residual-free bubbles finite element method. Multiscale Model. Simul. 1(3), 485–503 (2003) (electronic)Google Scholar
  32. 32.
    Wikipedia, Embarrassingly parallel (2013)Google Scholar

Copyright information

© Springer-Verlag Italia 2014

Authors and Affiliations

  1. 1.Laboratório Nacional de Computação CientíficaPetrópolis-RJBrazil
  2. 2.Fundação getúlio vargasRio de Janeiro-RJBrazil

Personalised recommendations