Skip to main content
Springer Nature Link
Log in

A Mortar Mixed Finite Volume Method for Elliptic Problems on Non-matching Multi-block Triangular Grids

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

Abstract

A mixed finite volume method is considered for the mixed formulation of second-order elliptic equations. The computational domain can be decomposed into non-overlapping sub-domains or blocks and the diffusion tensors may be discontinuous across the sub-domain boundaries. We define a conforming triangular partition on each sub-domain independently, and employ the standard mixed finite volume method within each sub-domain. A mortar finite element space is introduced to approximate the trace of the pressure on the non-matching interfaces. Moreover, a continuity condition of flux is imposed weakly. We prove the scheme’s first order optimal rate of convergence for both the pressure and the velocity. Numerical experiments are provided to illustrate the error behavior of the scheme and confirm our theoretical results.

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

Similar content being viewed by others

References

  1. Arbogast, T., Cowsar, L.C., Wheeler, M.F., Yotov, I.: Mixed finite element methods on non-matching multiblock grids. SIAM J. Numer. Anal. 37, 1295–1315 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. Arbogast, T., Xiao, H.L.: Two-level mortar domain decomposition preconditioners for heterogeneous elliptic problems. Comput. Methods Appl. Mech. Eng. 292, 221–242 (2015)

    Article  MathSciNet  Google Scholar 

  3. Arbogast, T., Yotov, I.: A non-mortar mixed finite element method for elliptic problems on non-matching multiblock grids. Comput. Methods Appl. Mech. Eng. 149, 255–265 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bi, C.J.: Superconvergence of mixed covolume method for elliptic problems on triangular grids. J. Comput. Appl. Math. 216, 534–544 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, New York (1991)

    Book  MATH  Google Scholar 

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

  7. Cai, Z.Q.: On the finite volume element method. Numer. Math. 58, 713–735 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chou, S.H., Kwak, D.Y.: Mixed covolume methods on rectangular grids for elliptic problems. SIAM J. Numer. Anal. 37, 758–771 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chou, S.H., Kwak, D.Y., Kim, K.Y.: A general framework for constructing and analyzing mixed finite volume methods on quadrilateral grids: the overlapping covolume case. SIAM J. Numer. Anal. 39, 1170–1196 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chou, S.H., Kwak, D.Y., Kim, K.Y.: Mixed finite volume methods on nonstaggered quadrilateral grids for elliptic problems. Math. Comput. 72, 525–539 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chou, S.H., Kwak, D.Y., Vassilevski, P.S.: Mixed covolume methods for elliptic problems on triangular grids. SIAM J. Numer. Anal. 35, 1850–1861 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chou, S.H., Li, Q.: Error estimates in \(L^2\), \(H^1\), \(L^\infty \) in covolume methods for elliptic and parabolic problem: a unified approach. Math. Comput. 69, 103–120 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  13. Ciarlet, P.G., Raviart, P.A., Lions, J.L.: General lagrange and hermite interpolation in \({\mathbb{R}}^n\) with alpplications to finite element methods. Arch. Ration. Mech. Anal. 46, 177–199 (1972)

    Article  MATH  Google Scholar 

  14. Chen, Z.Y., Xu, Y.S., Zhang, Y.Y.: A construction of highter-order finite volume methods. Math. Comput. 84(292), 599–628 (2014)

    Article  Google Scholar 

  15. Ganis, B., Wheeler, M.F., Yotov, I.: An enhanced velocity multipoint flux mixed finite element method for Darcy flow on non-matching hexahedral grids. Proced. Comput. Sci. 51, 1198–1207 (2015)

    Article  Google Scholar 

  16. Girault, V., Vassilev, D., Yotov, I.: Mortar multiscale finite element methods for Stokes–Darcy flows. Numer. Math. 127, 93–165 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. Li, R.H., Chen, Z.Y., Wu, W.: Generalized Difference Methods for Differential Equations: Numerical Analysis of Finite Volume Methods. Marcel Dekker, New York (2000)

    MATH  Google Scholar 

  18. Lv, J.L., Li, Y.H.: Optimal biquadratic finite volume element methods on quadrilateral meshes. SIAM J. Numer. Anal. 50, 2379–2399 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  19. Mishev, I.D.: Analysis of a new mixed finite volume method. Comput. Methods Appl. Math. 3, 189–201 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  20. Pan, H., Rui, H.X.: Mixed element method for two-dimensional Darcy–Forchheimer model. J. Sci. Comput. 52, 563–587 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. Rui, H.X.: Superconvergence of a mixed covolume method for elliptic problems. Computing 71, 247–263 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  22. Rui, H.X.: Convergence of an upwind control-volume mixed finite element method for convection-diffusion problems. Computing 81, 297–315 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  23. Rui, H.X., Lu, T.C.: An expanded mixed covolume method for elliptic problems. Numer. Methods Partial Differ. Equ. 21, 8–23 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  24. Rui, H.X., Zhang, R.: A unified stabilized mixed finite element method for coupling Stokes and Darcy flows. Comput. Methods Appl. Mech. Eng. 198, 2692–2699 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  25. Thomas, S.G., Wheeler, M.F.: Enhanced velocity mixed finite element methods for modeling coupled flow and transport on non-matching multiblock grids. Comput. Geosci. 15, 605–625 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  26. Tian, W.F., Li, Y.H.: Superconvergence of mixed covolume method on quadrilateral grids for elliptic problems. J. Syst. Sci. Complex. 25, 385–397 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  27. Tian, W.F., Song, L.Q., Li, Y.H.: A stabilized equal-order finite volume method for the stokes equations. J. Comput. Math. 30, 615–628 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  28. Tunc, X., Faille, I., Gallouët, T., Cacas, M.C., Havé, P.: A model for conductive faults with non-matching grids. Comput. Geosci. 16, 277–296 (2012)

    Article  MATH  Google Scholar 

  29. Vassilev, D., Wang, C.Q., Yotov, I.: Domain decomposition for coupled Stokes and Darcy flows. Comput. Methods Appl. Mech. Eng. 268, 264–283 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  30. Wheeler, J.A., Wheeler, M.F., Yotov, I.: Enhanced velocity mixed finite element methods for flow in multiblock domains. Comput. Geosci. 6, 315–332 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  31. Yotov, I.: Mixed finite element methods for flow in porous media, Ph.D. Thesis, Rice University, Houston (1996)

  32. Zhao, X.K., Chen, Y.L., Lv, J.L.: A mixed nonoverlapping covolume method on quadrilateral grids for elliptic problems. J. Comput. Appl. Math. 292, 23–40 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  33. Zhang, Z.M., Zou, Q.S.: Vertex-centered finite volume schemes of any order over quadrilateral meshes for elliptic boundary value problems. Numer. Math. 130, 363–393 (2015)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We appreciate the anonymous referees for their constructive comments on improving this paper.

Author information

Authors and Affiliations

  1. College of Mathematics, Jilin University, Changchun, 130012, Jilin, People’s Republic of China

    Yanni Gao & Yonghai Li

Authors
  1. Yanni Gao
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Yonghai Li
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Yonghai Li.

Additional information

This work is partially supported by the National Natural Science Foundation of China under Grant Number 11371170, the NASF under Grant Number U1630249 and the Science Challenge Program under Number JCKY2016212A502.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gao, Y., Li, Y. A Mortar Mixed Finite Volume Method for Elliptic Problems on Non-matching Multi-block Triangular Grids. J Sci Comput 73, 50–69 (2017). https://doi.org/10.1007/s10915-017-0405-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-017-0405-z

Keywords

Mathematics Subject Classification