Skip to main content
Log in

Convergence of Adaptive Weak Galerkin Finite Element Methods for Second Order Elliptic Problems

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

Abstract

We consider a standard Adaptive weak Galerkin (AWG) finite element method for second order elliptic problems. We prove that the sum of the energy error and the scaled error estimator of AWG method, between two consecutive adaptive loops, is a contraction. At last, we present some numerical experiments to support the theoretical results.

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

Access this article

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

Similar content being viewed by others

References

  1. Binev, P., Dahmen, W., DeVore, R.A.: Adaptive finite element methods with convergence rates. Numer. Math. 97(2), 219–268 (2004)

    Article  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

  3. Cascon, J.M., Kreuzer, C., Nochetto, R.H., Siebert, K.G.: Quasi-optimal convergence rate for an adaptive finite element method. SIAM J. Numer. Anal. 46(5), 2524–2550 (2008)

    Article  MathSciNet  Google Scholar 

  4. Chen, L.: iFEM: An Integrated Finite Element Methods Package in MATLAB. University of California, Irvine (2009)

    Google Scholar 

  5. Chen, L., Holst, M., Xu, J.: Convergence and optimality of adaptive mixed finite element methods. Math. Comput. 78(265), 35–53 (2009)

    Article  MathSciNet  Google Scholar 

  6. Chen, L., Wang, J., Ye, X.: A posteriori error estimates for weak Galerkin finite element methods for second order elliptic problems. J. Sci. Comput. 59(2), 496–511 (2014)

    Article  MathSciNet  Google Scholar 

  7. Du, Y., Zhang, Z.: A numerical analysis of the weak Galerkin method for the Helmholtz equation with high wave number. Commun. Comput. Phys. 22(1), 133–156 (2017)

    Article  MathSciNet  Google Scholar 

  8. Dörfler, W.: A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal. 33(3), 1106–1124 (1996)

    Article  MathSciNet  Google Scholar 

  9. Huang, J., Xu, Y.: Convergence and complexity of arbitrary order adaptive mixed element methods for the Poisson equation. Sci. China Math. 55(5), 1083–1098 (2012)

    Article  MathSciNet  Google Scholar 

  10. Kellogg, R.B.: On the Poisson equation with intersecting interfaces. Appl. Anal. 4(2), 101–129 (1974)

    Article  MathSciNet  Google Scholar 

  11. Li, H.G., Mu, L., Ye, X.: A posteriori error estimates for the weak Galerkin finite element methods on polytopal meshes. Commun. Comput. Phys. 26(2), 558–578 (2019)

    Article  MathSciNet  Google Scholar 

  12. Lin, G., Liu, J., Mu, L., Ye, X.: Weak Galerkin finite element methods for Darcy flow: anisotropy and heterogeneity. J. Comput. Phys. 276, 422–437 (2014)

    Article  MathSciNet  Google Scholar 

  13. Mitchell, W.F.: A comparison of adaptive refinement techniques for elliptic problems. ACM Trans. Math. Softw. 15(4), 326–347 (1989)

    Article  MathSciNet  Google Scholar 

  14. Morin, P., Nochetto, R.H., Siebert, K.G.: Data oscillation and convergence of adaptive FEM. SIAM J. Numer. Anal. 38(2), 466–488 (2000)

    Article  MathSciNet  Google Scholar 

  15. Morin, P., Nochetto, R.H., Siebert, K.G.: Convergence of adaptive finite element methods. SIAM Rev. 44(4), 631–658 (2002)

    Article  MathSciNet  Google Scholar 

  16. Mu, L.: Weak Galerkin based a posteriori error estimates for second order elliptic interface problems on polygonal meshes. J. Comput. Appl. Math. 361, 413–425 (2019)

    Article  MathSciNet  Google Scholar 

  17. Mu, L., Wang, J., Wei, G., Zhao, S.: Weak Galerkin methods for second order elliptic interface problems. J. Comput. Phys. 250, 106–125 (2013)

    Article  MathSciNet  Google Scholar 

  18. Mu, L., Wang, J., Ye, X.: Weak Galerkin finite element methods for the biharmonic equation on polytopal meshes. Numer. Methods Part. Differ. Equ. 30(3), 1003–1029 (2014)

    Article  MathSciNet  Google Scholar 

  19. Mu, L., Wang, J., Ye, X.: A new weak Galerkin finite element method for the Helmholtz equation. IMA J. Numer. Anal. 35(3), 1228–1255 (2015)

    Article  MathSciNet  Google Scholar 

  20. Mu, L., Wang, J., Ye, X.: A weak Galerkin finite element method with polynomial reduction. J. Comp. Appl. Math. 285, 45–58 (2015)

    Article  MathSciNet  Google Scholar 

  21. Mu, L., Wang, J., Ye, X.: Weak Galerkin finite element methods on polytopal meshes. Int. J. Numer. Anal. Mod. 12(1), 31–53 (2015)

    MathSciNet  MATH  Google Scholar 

  22. Mu, L., Wang, J., Ye, X., Zhang, S.: A \(C^{0}\)-weak Galerkin finite element method for the biharmonic equation. J. Sci. Comput. 59(2), 473–495 (2014)

    Article  MathSciNet  Google Scholar 

  23. Mu, L., Wang, J., Ye, X., Zhao, S.: A numerical study on the weak Galerkin method for the Helmholtz equation. Commun. Comput. Phys. 15(5), 1461–1479 (2014)

    Article  MathSciNet  Google Scholar 

  24. Nochetto, R.H., Siebert, K.G., Veeser, A.: Theory of adaptive finite element methods: an introduction. In: Devore, R.A., Kunoth, A. (eds.) Multiscale, Nonlinear and Adaptive Approximation, pp. 409–542. Springer, Berlin (2009)

    Chapter  Google Scholar 

  25. Stevenson, R.: The completion of locally refined simplicial partitions created by bisection. Math. Comp. 77(261), 227–241 (2008)

    Article  MathSciNet  Google Scholar 

  26. Wang, C., Wang, J.: An efficient numerical scheme for the biharmonic equation by weak Galerkin finite element methods on polygonal or polyhedral meshes. Comput. Math. Appl. 68(12), 2314–2330 (2013)

    Article  MathSciNet  Google Scholar 

  27. Wang, J., Ye, X.: A weak Galerkin finite element method for second-order elliptic problems. J. Comput. Appl. Math. 241(0), 103–115 (2013)

    Article  MathSciNet  Google Scholar 

  28. Wang, J., Ye, X.: A weak Galerkin mixed finite element method for second-order elliptic problems. Math. Comput. 83(289), 2101–2126 (2014)

    Article  MathSciNet  Google Scholar 

  29. Zhang, T., Chen, Y.: A posteriori error analysis for the weak Galerkin method for solving elliptic problems. Int. J. Comput. Methods 15(8), 1850075 (2018)

    Article  MathSciNet  Google Scholar 

  30. Zhang, J., Zhang, K., Li, J., Wang, X.: A weak Galerkin finite element method for the Navier–Stokes equations. Commun. Comput. Phys. 23, 706–746 (2018)

    MathSciNet  Google Scholar 

  31. Zhang, T., Lin, T.: A posteriori error estimate for a modified weak Galerkin method solving elliptic problems. Numer. Methods Part. D. E 33(1), 381–398 (2017)

    Article  MathSciNet  Google Scholar 

  32. Zheng, X., Xie, X.: A posteriori error estimator for a weak Galerkin finite element solution of the Stokes problem. E. Asian. J. Appl. Math. 7(3), 508–529 (2017)

    Article  MathSciNet  Google Scholar 

  33. Zhong, L., Chen, L., Shu, S., Wittum, G., Xu, J.: Convergence and optimality of adaptive edge finite element methods for time-harmonic Maxwell equations. Math. Comput. 81(278), 623–642 (2012)

    Article  MathSciNet  Google Scholar 

  34. Zhong, L., Shu, S., Chen, L., Xu, J.: Convergence of adaptive edge finite element methods for H(curl)-elliptic problems. Numer. Linear Algebra Appl. 17(2–3), 415–432 (2010)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank Professor Long Chen, University of California at Irvine, for providing many constructive suggestions.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Liuqiang Zhong.

Additional information

Publisher's Note

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

This research was supported in part by supported by the National Natural Science Foundation of China (Nos. 11671159, 12071160), the Guangdong Basic and Applied Basic Research Foundation (No. 2019A1515010724), the Characteristic Innovation Projects of Guangdong colleges and universities, China (No. 2018KTSCX044) and the General Project topic of Science and Technology in Guangzhou, China (No. 201904010117).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xie, Y., Zhong, L. Convergence of Adaptive Weak Galerkin Finite Element Methods for Second Order Elliptic Problems. J Sci Comput 86, 17 (2021). https://doi.org/10.1007/s10915-020-01387-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-020-01387-7

Keywords

Mathematics Subject Classification

Navigation