Skip to main content
SpringerLink
Log in

Superconvergence in H1-norm of a difference finite element method for the heat equation in a 3D spatial domain with almost-uniform mesh

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

In this paper, we propose a novel difference finite element (DFE) method based on the P1-element for the 3D heat equation on a 3D bounded domain. One of the novel ideas of this paper is to use the second-order backward difference formula (BDF) combining DFE method to overcome the computational complexity of conventional finite element (FE) method for the high-dimensional parabolic problem. First, we design a fully discrete difference FE solution \({u^{n}_{h}}\) by the second-order backward difference formula in the temporal t-direction, the center difference scheme in the spatial z-direction, and the P1-element on a almost-uniform mesh Jh in the spatial (x, y)-direction. Next, the H1-stability of \({u_{h}^{n}}\) and the second-order H1-convergence of the interpolation post-processing function on \({u_{h}^{n}}\) with respect to u(tn) are provided. Finally, numerical tests are presented to show the second-order H1-convergence results of the proposed DFE method for the heat equation in a 3D spatial domain.

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
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  1. Arbogast, T., Dawson, C.N., Keenan, P.T., Wheeler, M.F., Yotov, I.: Enhanced cell-centered finite differences for elliptic equations on general geometry. SIAM J. Numer. Anal. 19, 404–425 (1998)

    MathSciNet  Google Scholar 

  2. Arbogast, T., Wheeler, M.F., Yotov, I.: Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite differences. SIAM J. Numer. Anal. 34, 828–852 (1997)

    Article  MathSciNet  Google Scholar 

  3. Brandts, J., Krizek, M.: History and future of super-convergence in three-dimensional finite element methods. In: Preceedings of the Conference on Finite Element Methods: Three-dimensional Problems, GAKUTO Internat, Ser. Math. Sci. Appl. 15, Gakkotosho, Tokyo, pp 24–35 (2001)

  4. Bank, R. E., Xu, J.: Asymptotic exact a posteriori error estimators, Part I: Grids with superconvergence. SIAM J. Numer. Anal. 41, 2294–2312 (2003)

    Article  MathSciNet  Google Scholar 

  5. Bank, R. E., Xu, J.: Asymptotic exact a posteriori error estimators, Part II: General unstructured grids. SIAM J. Numer. Anal. 41, 2313–2332 (2003)

    Article  MathSciNet  Google Scholar 

  6. Bank, R. E., Xu, J., Zheng, B.: Superconvergent derivative recovery for Lagrange triangular elements of degree p on unstructured grids. SIAM J. Numer. Anal. 45, 2032–2046 (2007)

    Article  MathSciNet  Google Scholar 

  7. Baranger, J., Maitre, J. F., Oudin, F.: Connection between finite volume and mixed finite element methods. Modél Math. Anal. Numér. 30(4), 445–465 (1996)

    Article  MathSciNet  Google Scholar 

  8. Barbeiro, S.: Supraconvergent cell-centered scheme for two dimensional elliptic problems. Appl. Numer. Math. 59(1), 56–72 (2009)

    Article  MathSciNet  Google Scholar 

  9. Barbeiro, S., Ferreira, J. A., Grigorieff, R. D.: Supraconvergence of a finite difference scheme in hs(0, L). IMA J. Numer. Anal. 25, 797–811 (2005)

    Article  MathSciNet  Google Scholar 

  10. Ben-Artzi, M., Croisille, J. P., Fishelov, D.: Convergence of a compact scheme for the pure streamfunction formulation of the unsteady Navier-Stokes equations. SIAM J. Numer. Anal. 44, 1997–2024 (2006)

    Article  MathSciNet  Google Scholar 

  11. Brenner, S. C., Scott, L. R.: The Mathematical Theory of Finite Element Methods. Springer, New York and Berlin (2002)

    Book  Google Scholar 

  12. Babuska, I., Strouboulis, T., Upadhyay, C. S., Gangaraj, K.: Computer-based proof of the existence of superconvergence points in the finite element methods; superconvergence of the derivatives in finite element solutions of Laplace’s, Poisson’s and elasticity equations. Numer. Meth Part. D. E. 12, 347–392 (1996)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  14. Cai, Z., Douglas, J. Jr, Park, M.: Development and analysis of higher order finite volume methods over rectangles for elliptic equations. Adv. Comput. Math. 3-33, 19 (2003)

    MathSciNet  Google Scholar 

  15. Cai, Z., Mandel, J., McCormick, S. F.: The finite volume element method for diffusion equations on general triangulations. SIAM J. Numer. Anal. 392-402, 28 (1991)

    MathSciNet  Google Scholar 

  16. Chen, C.: Derivative superconvergence points for triangular element approximation in FEM. J. Xiangtan Univ. 1, 77–90 (1978)

    Google Scholar 

  17. Chen, C.: Derivative superconvergence points for linear triangular finite element. Numer. Math. J. Chinese Univ. 2, 12–20 (1980)

    MathSciNet  Google Scholar 

  18. Chen, C., Huang, Y.: High Accuracy Analysis of the Finite Element Method. Hunan Academic Tecnique Press, Changsha (1995). (in Chinese)

    Google Scholar 

  19. Chen, C.: Structure Theory of Superconvergence of Finite Elements. Hunan Academic Press, Changsha (2001). (in Chinese)

    Google Scholar 

  20. Chen, Z.: Finite Element Methods and their Applications. Springer, Heidelberg and New York (2005)

    Google Scholar 

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

    Google Scholar 

  22. Feng, X., He, R., Chen, Z.: H1-superconvergence of finite difference method based on Q1-element on quasi-uniform mesh for the 3D Poisson equation. Numer. Meth. Part. D. E. 36 (1), 29–48 (2020)

    Article  Google Scholar 

  23. Feng, X., He, Y.: H1-Super-convergence of center finite difference method based on P1-element for the elliptic equation. Appl. Math. Model. 58, 5439–5453 (2014)

    Article  Google Scholar 

  24. Feng, X., Li, R., He, Y., Liu, D.: P1-nonconforming quadrilateral finite volume methods for the semilinear elliptic equations. J. Sci. Comput. 52(3), 519–545 (2012)

    Article  MathSciNet  Google Scholar 

  25. Feng, X., Tang, T., Yang, J.: Long time numerical simulations for phase-field problems using p-adaptive spectral deferred correction methods. SIAM J. Sci. Comput. 37(1), A271–A294 (2015)

    Article  MathSciNet  Google Scholar 

  26. Ferreira, J.A.: Supraconvergence of elliptic finite difference schemes: general boundary conditions and low regularity, Pré-publicações do Departamento de Matemática da Universidade de Coimbra 04-27 (2004)

  27. Ferreira, J. A., Grigorieff, R. D.: On the supraconvergence of elliptic finite difference schemes. Appl. Numer. Math. 275–292, 28 (1998)

    MathSciNet  Google Scholar 

  28. Ferreira, J. A., Grigorieff, R. D.: Supraconvergence and supercloseness of a scheme for elliptic equations on nonuniform grids. Numer. Func. Anal. Opt. 27(5-6), 539–564 (2006)

    Article  MathSciNet  Google Scholar 

  29. Forsyth, P.A., Sammon, P.H.: Quadratic convergence for cell-centered grids. Appl. Numer. Math. 4, 377–394 (1988)

    Article  MathSciNet  Google Scholar 

  30. He, Y., Feng, X.: H1-Stability and convergence of the FE, FV and FD methods for an elliptic equation. East Asian J. Appl. Math. 3(2), 154–170 (2013)

    Article  MathSciNet  Google Scholar 

  31. He, R., Feng, X.: Second order convergence of the interpolation based on \({q_{1}^{c}}\)-element. Numer. Math. Theory, Methods & Appl. 9(4), 595–618 (2016)

    Article  MathSciNet  Google Scholar 

  32. He, R., Feng, X., Chen, Z.: H1-superconvergence of a difference finite element method based on the p1 − p1-conforming element on non-uniform meshes for the 3D Poisson equation. Math. Comp. 87, 1659–1688 (2018)

    Article  MathSciNet  Google Scholar 

  33. Huan, Y., Xu, J.: Superconvergence of quadratic finite elements on mildly structured grids. Math. Comp. 77, 1253–1268 (2008)

    Article  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

  35. Lin, R., Zhang, Z.: Natural superconvergence points in three-dimensional finite element. SIAM J. Numer. Anal. 46, 1281–1297 (2008)

    Article  MathSciNet  Google Scholar 

  36. Lin, Q., Yan, N.: Construction and Analysis of High Efficient Finite Element. Heibei University Press, Baoding (1996). (in Chinese)

    Google Scholar 

  37. Manteuffel, T. A., White, A.B. Jr: The numerical solution of second order boundary value problems on nonuniform meshes. Math. Comp. 47, 511–535 (1986)

    Article  MathSciNet  Google Scholar 

  38. Nakata, M., Weiser, A., Wheeler, M. F.: Some superconvergence results for mixed finite element methods for elliptic problems on rectangular domains, Technical Report, Rice University and Exxon Production Research Co., Houston, TX (1984)

  39. Rannacher, R., Scott, R.: Some optimal error estimates for piecewise linear finite element approximation. Math. Comp. 38, 437–445 (1982)

    Article  MathSciNet  Google Scholar 

  40. Russell, T. F., Wheeler, M. F.: Finite element and finite difference methods for continuous flows in porous media. In: Ewing, R.E. (ed.) The Mathematics of Reservoir Simulation. SIAM, Philadelphia (1983)

  41. Samarskii, A. A., Lazarov, R. D., Makarov, L.: Finite Difference Schemes for Differential Equations with Weak Solutions. Moscow Vissaya Skola, Moscow (1987)

    Google Scholar 

  42. Schatz, A. H., Sloan, I. H., Wahlbin, L. R.: Super-convergence in finite element methods and meshes that are locally symmetric with respect to a point. SIAM J. Numer. Anal. 33, 505–521 (1996)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  44. Shi, D., Liang, H.: The superconvergence analysis of linear triangular element on anisotropic meshes. Chinese J. Eng. Math. 24, 487–493 (2007)

    MathSciNet  Google Scholar 

  45. Thomas, J. W.: Numerical Partial Differential Equations? Conservation Laws and Elliptic Equations. Springer, New York (1999)

    Book  Google Scholar 

  46. Thomas, J. W.: Numerical Partial Differential Equations—Finite Difference Methods. Springer, New York (1998)

    Google Scholar 

  47. Weiser, A., Wheeler, M. F.: On convergence of block-centered finite differences for elliptic problems. SIAM J. Numer. Anal. 351-375, 25 (1988)

    MathSciNet  Google Scholar 

  48. Xu, J.: Two-grid discretization techniques for linear and nonlinear PDEs. SIAM J. Numer. Anal. 33, 1759–1777 (1996)

    Article  MathSciNet  Google Scholar 

  49. Xu, J., Zou, Q.: Analysis of linear and quadratic simplicial finite volume methods for elliptic equations. Numer. Math. 111, 469–492 (2009)

    Article  MathSciNet  Google Scholar 

  50. Yu, H., Huang, Y.: Asymptotic expansion and superconvergence for triangular linear finite element on a class of typical mesh. Int. J. Numer. Anal Model. 9, 892–906 (2012)

    MathSciNet  Google Scholar 

  51. Zhai, S., Feng, X., He, Y.: A family of fourth-order and sixth-order compact difference schemes for the three-dimensional Poisson equation. J. Sci. Comput. 54 (1), 97–120 (2013)

    Article  MathSciNet  Google Scholar 

  52. Zhai, S., Feng, X., He, Y.: A new method to deduce high-order compact difference schemes for two-dimensional Poisson equation. Appl. Math. Comput. 230, 9–26 (2014)

    MathSciNet  Google Scholar 

  53. Zhai, S., Feng, X., He, Y.: Numerical simulation of the three dimensional Allen-Cahn equation by the high-order compact ADI method. Comput. Phys. Commun. 185(10), 2449–2455 (2014)

    Article  Google Scholar 

  54. Zhang, Z.: Derivative superconvergence points in finite element solutions of Poisson equation for the serendipity and intermediate families-a theory justification. Math. Comp. 67, 541–552 (1998)

    Article  MathSciNet  Google Scholar 

  55. Zhang, Z.: Derivative superconvergence points in finite element points of harmonic functions-a theory justification. Math. Comp. 71, 1421–1430 (2002)

    Article  MathSciNet  Google Scholar 

  56. Zhang, Z., Lin, R.: Localting natural superconvergence points of finite element methods in 3D. Int. J. Numer. Anal. Model. 2, 19–30 (2005)

    MathSciNet  Google Scholar 

  57. Zhu, Q.: Superconvergence of derivative for quadratic triangular finite element. J. Xiangtan Univ. 3, 36–45 (1981)

    Google Scholar 

  58. Zhu, Q., Lin, Q.: Finite Element Superconvergence Theory. Hunan Science Press, Changsha (1989). (in Chinese)

    Google Scholar 

Download references

Acknowledgments

The authors would like to thank the editor and referees for their valuable comments and suggestions which helped us to improve the results of this article.

Funding

This research has been made possible by contributions from the NSF of China (No. U19A2079, No. 11671345, and No. 11362021), the Xinjiang Provincial University Research Foundation of China, NSERC/AIEES/Foundation CMG IRC in Reservoir Simulation, AITF (iCore) Chair in Reservoir Modelling, and the Frank and Sarah Meyer Foundation CMG Collaboration Centre.

Author information

Authors and Affiliations

  1. College of Mathematics and Systems Science, Xinjiang University, Urumqi, 830046, People’s Republic of China

    Xinlong Feng

  2. Department of Chemical and Petroleum Engineering, Schulich School of Engineering, University of Calgary, Calgary, AB, T2N 1N4, Canada

    Ruijian He & Zhangxin Chen

Authors
  1. Xinlong Feng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ruijian He
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zhangxin Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xinlong Feng.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Feng, X., He, R. & Chen, Z. Superconvergence in H1-norm of a difference finite element method for the heat equation in a 3D spatial domain with almost-uniform mesh. Numer Algor 86, 357–395 (2021). https://doi.org/10.1007/s11075-020-00892-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-020-00892-y

Keywords

Mathematics Subject Classification (2010)

Search

Navigation