Skip to main content
SpringerLink
Log in

Multigrid methods for saddle point problems: Darcy systems

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

We design and analyze multigrid methods for the saddle point problems resulting from Raviart–Thomas–Nédélec mixed finite element methods for the Darcy system in porous media flow. Uniform convergence of the W-cycle algorithm in a nonstandard energy norm is established. Extensions to general second order elliptic problems are also addressed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

References

  1. Allen, M.B., Ewing, R.E., Lu, P.: Well-conditioned iterative schemes for mixed finite-element models of porous-media flows. SIAM J. Sci. Stat. Comput. 13, 794–814 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  2. Antonietti, P.F., Ayuso de Dios, B.: Schwarz domain decomposition preconditioners for discontinuous Galerkin approximations of elliptic problems: non-overlapping case. M2AN Math. Model. Numer. Anal. 41, 21–54 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Arnold, D.N., Brezzi, F.: Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates. RAIRO Modél. Math. Anal. Numér. 19, 7–32 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  4. Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39, 1749–1779 (2001/02)

  5. Babuška, I., Osborn, J., Pitkäranta, J.: Analysis of mixed methods using mesh dependent norms. Math. Comput. 35, 1039–1062 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  6. Babuška, I.: The finite element method with Lagrange multipliers. Numer. Math. 20, 179–192 (1973)

    Article  MATH  Google Scholar 

  7. Bank, R.E., Dupont, T.F.: An optimal order process for solving finite element equations. Math. Comput. 36, 35–51 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ben Belgacem, F., Brenner, S.C.: Some nonstandard finite element estimates with applications to 3D Poisson and Signorini problems. Electron. Trans. Numer. Anal. 12, 134–148 (2001)

    MathSciNet  MATH  Google Scholar 

  9. Benzi, M., Golub, G.H., Liesen, J.: Numerical solution of saddle point problems. Acta Numer. 14, 1–137 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. Boffi, D., Brezzi, F., Fortin, M.: Mixed Finite Element Methods and Applications. Springer, Heidelberg (2013)

    Book  MATH  Google Scholar 

  11. Bramble, J.H., Hilbert, S.R.: Estimation of linear functionals on Sobolev spaces with applications to Fourier transforms and spline interpolation. SIAM J. Numer. Anal. 7, 113–124 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bramble, J.H., Xu, J.: Some estimates for a weighted \(L^2\) projection. Math. Comput. 56, 463–476 (1991)

    MathSciNet  MATH  Google Scholar 

  13. Bramble, J.H., Zhang, X.: The analysis of multigrid methods. In: Ciarlet, P.G., Lions, J.L. (eds.) Handbook of Numerical Analysis, vol. VII, pp. 173–415. North-Holland, Amsterdam (2000)

    Google Scholar 

  14. Brenner, S.C.: A multigrid algorithm for the lowest-order Raviart–Thomas mixed triangular finite element method. SIAM J. Numer. Anal. 29, 647–678 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  15. Brenner, S.C.: Convergence of nonconforming multigrid methods without full elliptic regularity. Math. Comput. 68, 25–53 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  16. Brenner, S.C.: Poincaré-Friedrichs inequalities for piecewise \(H^1\) functions. SIAM J. Numer. Anal. 41, 306–324 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  17. Brenner, S.C., Li, H., Sung, L.-Y.: Multigrid methods for saddle point problems: Stokes and Lamé systems. Numer. Math. 128, 193–216 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, 3rd edn. Springer, New York (2008)

    Book  MATH  Google Scholar 

  19. Brenner, S.C., Zhao, J.: Convergence of multigrid algorithms for interior penalty methods. Appl. Numer. Anal. Comput. Math. 2, 3–18 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  20. Brezzi, F.: On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. RAIRO Anal. Numér. 8, 129–151 (1974)

    MathSciNet  MATH  Google Scholar 

  21. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)

    MATH  Google Scholar 

  22. Crouzeix, M., Raviart, P.-A.: Conforming and nonconforming finite element methods for solving the stationary Stokes equations I. RAIRO Anal. Numér. 7, 33–75 (1973)

    MathSciNet  MATH  Google Scholar 

  23. Dauge, M.: Elliptic Boundary Value Problems on Corner Domains. Lecture Notes in Mathematics, vol. 1341. Springer, Berlin (1988)

    MATH  Google Scholar 

  24. Douglas Jr., J., Roberts, J.E.: Mixed finite element methods for second order elliptic problems. Math. Appl. Comput. 1, 91–103 (1982)

    MathSciNet  MATH  Google Scholar 

  25. Dupont, T., Scott, R.: Polynomial approximation of functions in Sobolev spaces. Math. Comput. 34, 441–463 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  26. Ewing, R., Wang, J.: Analysis of multilevel decomposition iterative methods for mixed finite element methods. M2AN Math. Model. Numer. Anal. 28, 377–398 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  27. Falk, R.S., Osborn, J.E.: Error estimates for mixed methods. RAIRO Anal. Numér. 14, 249–277 (1980)

    MathSciNet  MATH  Google Scholar 

  28. Feng, X., Karakashian, O.A.: Two-level additive Schwarz methods for a discontinuous Galerkin approximation of second order elliptic problems. SIAM J. Numer. Anal. 39, 1343–1365 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  29. Gopalakrishnan, J., Kanschat, G.: A multilevel discontinuous Galerkin method. Numer. Math. 95, 527–550 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  30. Grisvard, P.: Elliptic Problems in Non Smooth Domains. Pitman, Boston (1985)

    MATH  Google Scholar 

  31. Hackbusch, W.: Multi-grid Methods and Applications. Springer, Berlin (1985)

    Book  MATH  Google Scholar 

  32. Jaffré, J.: Décentrage et éléments finis mixtes pour les équations de diffusion-convection. Calcolo 21, 171–197 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  33. Kim, D., Park, E.-J.: A posteriori error estimators for the upstream weighting mixed methods for convection diffusion problems. Comput. Methods Appl. Mech. Eng. 197, 806–820 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  34. Lasser, C., Toselli, A.: An overlapping domain decomposition preconditioner for a class of discontinuous Galerkin approximations of advection-diffusion problems. Math. Comput. 72, 1215–1238 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  35. Lovadina, C., Stenberg, R.: Energy norm a posteriori error estimates for mixed finite element methods. Math. Comput. 75, 1659–1674 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  36. Mardal, K.-A., Winther, R.: Preconditioning discretizations of systems of partial differential equations. Numer. Linear Algebra Appl. 18, 1–40 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  37. Nazarov, S.A., Plamenevsky, B.A.: Elliptic Problems in Domains with Piecewise Smooth Boundaries. de Gruyter, Berlin (1994)

    Book  MATH  Google Scholar 

  38. Nédélec, J.-C.: Mixed finite elements in \({\mathbf{R}}^3\). Numer. Math. 35, 315–341 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  39. Raviart, P.-A., Thomas, J.M.: A mixed finite element method for second order elliptic problems. In: Galligani, I., Magenes, E. (eds.) Mathematical Aspects of the Finite Element Method. Lecture Notes in Mathematics, vol. 606, pp. 292–315. Springer, Berlin (1977)

    Chapter  Google Scholar 

  40. Rusten, T., Vassilevski, P.S., Winther, R.: Interior penalty preconditioners for mixed finite element approximations of elliptic problems. Math. Comput. 65, 447–466 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  41. Rusten, T., Winther, R.: A preconditioned iterative method for saddlepoint problems. SIAM J. Matrix Anal. Appl. 13, 887–904 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  42. Schatz, A.: An observation concerning Ritz–Galerkin methods with indefinite bilinear forms. Math. Comput. 28, 959–962 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  43. Stenberg, R.: On the construction of optimal mixed finite element methods for the linear elasticity problem. Numer. Math. 48, 447–462 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  44. Tartar, L.: An Introduction to Sobolev Spaces and Interpolation Spaces. Springer, Berlin (2007)

    MATH  Google Scholar 

  45. Vassilevski, P.S.: Multilevel Block Factorization Preconditioners. Springer, New York (2008)

    MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank Monique Dauge for helpful discussions on the regularity of the Darcy system. Portions of this research were conducted with high performance computational resources provided by Louisiana State University (http://www.hpc.lsu.edu).

Author information

Author notes

  1. Duk-Soon Oh

    Present address: Department of Mathematics, Rutgers University, Piscataway, NJ, 08854, USA

Authors and Affiliations

  1. Department of Mathematics and Center for Computation and Technology, Louisiana State University, Baton Rouge, LA, 70803, USA

    Susanne C. Brenner, Duk-Soon Oh & Li-Yeng Sung

Authors
  1. Susanne C. Brenner
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Duk-Soon Oh
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Li-Yeng Sung
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Susanne C. Brenner.

Additional information

The work of the authors Susanne C. Brenner and Li-Yeng Sung was supported in part by the National Science Foundation under Grant Numbers DMS-13-19172 and DMS-16-20273.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Brenner, S.C., Oh, DS. & Sung, LY. Multigrid methods for saddle point problems: Darcy systems. Numer. Math. 138, 437–471 (2018). https://doi.org/10.1007/s00211-017-0911-9

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-017-0911-9

Mathematics Subject Classification

Search

Navigation