Nothing Special   »   [go: up one dir, main page]

Skip to main content
Log in

Uniqueness of discrete solutions of nonmonotone PDEs without a globally fine mesh condition

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

Uniqueness of the finite element solution for nonmonotone quasilinear problems of elliptic type is established in one and two dimensions. In each case, we prove a comparison theorem based on locally bounding the variation of the discrete solution over each element. The uniqueness follows from this result, and does not require a globally small meshsize.

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

Access this article

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.

Similar content being viewed by others

References

  1. Abdulle, A., Vilmart, G.: A priori error estimates for finite element methods with numerical quadrature for nonmonotone nonlinear elliptic problems. Numer. Math. 121(3), 397–431 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alvino, A., Betta, M.F., Mercaldo, A.: Comparison principle for some classes of nonlinear elliptic equations. J. Differ. Equ. 249(12), 3279–3290 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. André, N., Chipot, M.: A remark on uniqueness for quasilinear elliptic equations. Banach Center Publ. 33(1), 9–18 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  4. André, N., Chipot, M.: Uniqueness and nonuniqueness for the approximation of quasilinear elliptic equations. SIAM J. Numer. Anal. 33(5), 1981–1994 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bi, C., Ginting, V.: A posteriori error estimates of discontinuous Galerkin method for nonmonotone quasi-linear elliptic problems. J. Sci. Comput. 55(3), 659–687 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Douglas, J., Dupont, T.: A Galerkin method for a nonlinear Dirichlet problem. Math. Comput. 131, 689–696 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  7. Douglas, J., Dupont, T., Serrin, J.: Uniqueness and comparison theorems for nonlinear elliptic equations in divergence form. Arch. Ration. Mech. Anal. 42(3), 157–168 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Grundlehren der mathematischen Wissenschaften: 224. Springer, Berlin (1983)

  9. Hlavác̆ek, I., Kr̆íz̆ek, M., Malý, J.: On Galerkin approximations of a quasilinear nonpotential elliptic problem of a nonmonotone type. J. Math. Anal. Appl. 184(1), 168–189 (1994)

    Article  MathSciNet  Google Scholar 

  10. Karátson, J., Korotov, S.: Discrete maximum principles for finite element solutions of nonlinear elliptic problems with mixed boundary conditions. Numer. Math. 99(4), 669–698 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Karátson, J., Korotov, S., Křížek, M.: On discrete maximum principles for nonlinear elliptic problems. Math. Comput. Simul. 76(1–3), 99–108 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Pollock, S.: A regularized Newton-like method for nonlinear PDE. Numer. Funct. Anal. Optim. 36(11), 1493–1511 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  13. Pollock, S.: An improved method for solving quasilinear convection diffusion problems. SIAM J. Sci. Comput. 38(2), A1121–A1145 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  14. Pollock, S.: Stabilized and inexact adaptive methods for capturing internal layers in quasilinear PDE. J. Comput. Appl. Math. 308, 243–262 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  15. Wang, J., Zhang, R.: Maximum principles for \(P1\)-conforming finite element approximations of quasi-linear second order elliptic equations. SIAM J. Numer. Anal. 50(2), 626–642 (2012)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to acknowledge an anonymous reviewer for feedback which helped to clarify several issues in this manuscript.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Sara Pollock.

Additional information

SP was supported in part by NSF DMS 1719849. YZ was supported in part by NSF DMS-1319110.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pollock, S., Zhu, Y. Uniqueness of discrete solutions of nonmonotone PDEs without a globally fine mesh condition. Numer. Math. 139, 845–865 (2018). https://doi.org/10.1007/s00211-018-0956-4

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-018-0956-4

Mathematics Subject Classification

Navigation