Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

A Unified Mortar Condition for Nonconforming Finite Elements

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

A continuously interpolated mortar condition is proposed for 2D and 3D \(P_k\) nonconforming finite elements on nonmatching grids. The resulting finite element method is an optimal order one in solving elliptic equations. Numerical tests on the 2D \(P_1\), 2D \(P_2\) and 3D \(P_1\) nonconforming finite elements are provided.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

References

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

    MathSciNet  MATH  Google Scholar 

  2. Babuška, I., Osborn, J.E.: Generalized finite element methods: their performance and their relation to mixed methods. SIAM J. Numer. Anal. 20(3), 510–536 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  3. Babuška, I., Caloz, G., Osborn, J.E.: Special finite element methods for a class of second order elliptic problems with rough coefficients. SIAM J. Numer. Anal. 31(4), 945–981 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bank, R.E., Xu, J.: An algorithm for coarsening unstructured meshes. Numer. Math. 73(1), 1–36 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  5. Baran, A., Stoyan, G.: Gauss-Legendre elements: a stable, higher order non-conforming finite element family. Computing 79, 1–21 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Belgacem, F.B.: The Mortar Element Method with Lagrange Multipliers. Université Paul Sabatier, Toulouse, France (1994)

    Google Scholar 

  7. Braess, D., Dahmen, W.: Stability estimates of the mortar finite element method for 3-dimensional problems. East-West J. Numer. Math. 6, 249–264 (1998)

    MathSciNet  MATH  Google Scholar 

  8. Braess, D., Dahmen, W.: The mortar element method revisited—what are the right norms? In Debit, N., Garbey, M., Hoppe, R., Périaux, J., Keyes, D., Kuznetsov, Y. (eds.) Proceedings of the Thirteenth International Conference on Domain Decomposition, pp. 245–252. Springer, Berlin (2001)

  9. Belgacem, F.B., Maday, Y.: The mortar element method for three dimensional finite elements. RAIRO Modél. Math. Nal. Numér. 31, 289–302 (1997)

    MATH  Google Scholar 

  10. Bernardi, C., Maday, Y., Patera, A.T.: A new non conforming approach to domain decomposition: the mortar element method. In: Brezis, H., Lions, J.-L. (eds.) Nonlinear Partial Differential Equations and Their Applications, Collége de France Seminar, pp. 13–51. Longman Scientific and Technical, Harlow, UK (1994)

    Google Scholar 

  11. Bertoluzza, S., Perrier, V.: The mortar method in the wavelet context. M2AN Math. Model Numer. Anal. 35, 647–674 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  12. Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, Texts in Applied Mathematics 15, 2nd edn. Springer, New York (2002)

    Book  Google Scholar 

  13. Chan, T.F., Xu, J., Zikatanov, L.: An agglomeration multigrid method for unstructured grids, Domain decomposition methods, 10 (Boulder, CO, 1997), 67–81, Contemp. Math., 218, Amer. Math. Soc., Providence, RI (1998)

  14. 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 

  15. Crouzeix, M., Raviart, P.-A.: Conforming and non-conforming finite elements for solving the stationary Stokes equations I. RAIRO Model Math. Anal. Numer. 7, 33–76 (1973). R-3

    MathSciNet  Google Scholar 

  16. Fortin, M., Soulie, M.: A nonconforming piecewise quadratic finite element on triangles. Int. J. Numer. Methods Eng. 19, 505–520 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hou, T.Y., Wu, X.-H.: A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys. 134(1), 169–189 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  18. Lacour, C., Maday, Y.: Two different approaches for matching nonconforming grids: the mortar element method and the FETI method. BIT 37(3), 720–738 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  19. Marcinkowski, L.: The mortar element method with locally nonconforming elements. BIT 39, 716–739 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  20. Marcinkowski, L.: Domain decomposition methods for mortar finite element discretizations of plate problems. SIAM J. Numer. Anal. 39, 1097–1114 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  21. Marcinkowski, L.: A mortar element method for some discretizations of a plate problem. Numer. Math. 93, 361–386 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  22. Marcinkowski, L.: An Additive Schwarz Method for mortar Morley finite element discretizations of 4th order elliptic problem in 2d. Electron. Trans. Numer. Anal. 26, 34–54 (2007)

    MathSciNet  MATH  Google Scholar 

  23. Marcinkowski, L.: A Neumann–Neumann algorithm for a mortar finite element discretization of fourth-order elliptic problems in 2D. Numer. Methods Partial Differ. Equ. 25, 1425–1442 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  24. Marcinkowski, L.: A preconditioner for a FETI-DP method for mortar element discretization of a 4th order problem in 2D. Electron. Trans. Numer. Anal. 38, 1–16 (2011)

    MathSciNet  MATH  Google Scholar 

  25. Marcinkowski, L., Rahman, T., Valdman, J.: A 3D Crouzeix-Raviart mortar finite element. Computing 86, 313–330 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  26. Mu, L., Wang, J., Ye, X., Zhang, S.: A \(C^0\)-weak Galerkin finite element method for the biharmonic equation. J. Sci. Comput. (accepted)

  27. Rahman, T., Bjørstad, P., Xu, X.: The Crouzeix-Raviart FE on nonmatching grids with an approximate method. SIAM J. Numer. Anal. 46, 496–516 (2008)

    Article  MATH  Google Scholar 

  28. Rahman, T., Xu, X.: A multilevel preconditioner for the mortar method for nonconforming P1 finite element. ESAIM-M2AN 43, 429–444 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  29. Rahman, T., Xu, X., Hoppe, R.: Additive Schwarz methods for the Crouzeix-Raviart mortar finite element for elliptic problems with discontinuous coefficients. Numer Math. 101, 551–572 (2003)

    Article  MathSciNet  Google Scholar 

  30. Sarkis, M.: Nonstandard coarse spaces and Schwarz methods for elliptic problems with discon-tinuous coefficients using non-conforming element. Numer. Math. 77, 383–406 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  31. Seshaiyer, P., Suri, M.: Uniform hp convergence results for the mortar finite element method. Math. Comput. 69, 521–546 (1999)

    Article  MathSciNet  Google Scholar 

  32. Scott, L., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54, 483–493 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  33. Wohlmuth, B.: A residual based error estimator for mortar finite element discretizations. Numer. Math. 84, 143–171 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  34. Wohlmuth, B.I.: A mortar finite element method using dual spaces for the Lagrange multiplier. SIAM J. Numer. Anal. 38, 989–1012 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  35. Xu, X., Chen, J.: Multigrid for the mortar element method for P1 nonconforming element. Numer. Math. 88, 381–398 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  36. Xu, X., Li, L., Chen, W.: A multigrid method for the mortar-type Morley element approximation of a plate bending problem. SIAM J. Numer. Anal. 39, 1721–1731 (2002)

    Article  MathSciNet  Google Scholar 

  37. Zhang, S.: Successive subdivisions of tetrahedra and multigrid methods on tetrahedral meshes. Houston J. Math. 21, 541–556 (1995)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The first and the third authors are supported in part by NSFC under the Grant 11071124 and 11371199. The first author is also supported in part by the Project of Graduate Education Innovation of Jiangsu Province under the Grant CXZZ13-0387. The third author is also supported in part by the Jiangsu Provincial 2011 Program (Collaborative Innovation Center of Climate Change).

Author information

Authors and Affiliations

  1. Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023, China

    Chunmei Wang & Jinru Chen

  2. School of Mathematical Sciences, Nanjing Normal University Taizhou College, Taizhou, 225300, China

    Chunmei Wang

  3. Department of Mathematical Sciences, University of Delaware, Newark, DE, 19716, USA

    Shangyou Zhang

Authors
  1. Chunmei Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shangyou Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jinru Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shangyou Zhang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, C., Zhang, S. & Chen, J. A Unified Mortar Condition for Nonconforming Finite Elements. J Sci Comput 62, 179–197 (2015). https://doi.org/10.1007/s10915-014-9852-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-014-9852-y

Keywords

Mathematics Subject Classification

Search

Navigation