Skip to main content
SpringerLink
Log in

Linearly implicit BDF methods for nonlinear parabolic interface problems

  • Published:
BIT Numerical Mathematics Aims and scope Submit manuscript

Abstract

We analyze linearly implicit BDF methods for the time discretization of a nonlinear parabolic interface problem, where the computational domain is divided into two subdomains by an interface, and the nonlinear diffusion coefficient is discontinuous across the interface. We prove optimal-order error estimates without assuming any growth conditions on the nonlinear diffusion coefficient and without restriction on the stepsize. Due to the existence of the interface and the lack of global Lipschitz continuity of the diffusion coefficient, we use a special type of test functions to analyze high-order \(A(\alpha )\)-stable BDF methods. Such test functions avoid any interface terms upon integration by parts and are used to derive error estimates in the piecewise \(H^2\) norm.

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

Similar content being viewed by others

References

  1. Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975)

    MATH  Google Scholar 

  2. Akrivis, G., Li, B., Lubich, C.: Combining maximal regularity and energy estimates for time discretizations of quasilinear parabolic equations. Math. Comp. doi:10.1090/mcom/3228

  3. Akrivis, G., Lubich, C.: Fully implicit, linearly implicit and implicit-explicit backward difference formulae for quasi-linear parabolic equations. Numer. Math. 131, 713–735 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  4. Babuška, I.: The finite element method for elliptic equations with discontinuous coefficients. Computing 5, 207–213 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  5. Boyer, F., Hubert, F.: Finite volume method for 2D linear and nonlinear elliptic problems with discontinuities. SIAM J. Numer. Anal. 46, 3032–3070 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bramble, J.H., King, J.T.: A finite element method for interface problems in domains with smooth boundaries and interfaces. Adv. Comput. Math. 6, 109–138 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cao, Y., Gunzburger, M., Hua, F., Wang, X.: Coupled Stokes–Darcy model with Beavers–Joseph interface boundary condition. Commun. Math. Sci. 8, 1–25 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chen, Z., Zou, J.: Finite element methods and their convergence for elliptic and parabolic interface problems. Numer. Math. 79, 175–202 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  9. Collisa, J.M., Siegmann, W.L., Jensen, F.B., Zampolli, M., Ksel, E.T., Collins, M.D.: Parabolic equation solution of seismo-acoustics problems involving variations in bathymetry and sediment thickness. J. Acoust. Soc. Am. 123, 51–55 (2008)

    Article  Google Scholar 

  10. Dahlquist, G.: G-stability is equivalent to A-stability. BIT 18, 384–401 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gao, H.: Unconditional optimal error estimates of BDF-Galerkin FEMs for nonlinear thermistor equations. J. Sci. Comput. 66, 504–527 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hairer, E., Wanner, G.: Solving ordinary differential equations II: stiff and differential–algebraic problems, In: Springer Series in Computational Mathematics, 2nd revised edn, vol. 14. Springer, Berlin (2002)

  13. Hansbo, A., Hansbo, P.: An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems. Comput. Methods Appl. Mech. Eng. 191, 537–552 (2002)

    MathSciNet  MATH  Google Scholar 

  14. Kellogg, R.B.: Singularities in interface problems. In: Hubbard, B. (ed.) Numerical Solution of Partial Differential Equations II, pp. 351–400. Academic Press, New York (1971)

    Chapter  Google Scholar 

  15. Li, B., Sun, W.: Error analysis of linearized semi-implicit Galerkin finite element methods for nonlinear parabolic equations. Int. J. Numer. Anal. Model. 10, 622–633 (2013)

    MathSciNet  MATH  Google Scholar 

  16. Li, B., Sun, W.: Unconditional convergence and optimal error estimates of a Galerkin-mixed FEM for incompressible miscible flow in porous media. SIAM J. Numer. Anal. 51, 1959–1977 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. Li, B., Rui, H., Yang, C.: Convergence of a decoupled mixed FEM for miscible displacement in interfacial porous media. arXiv:1406.2446

  18. Li, Z., Lin, T., Wu, X.: New Cartesian grid methods for interface problem using finite element formulation. Numer. Math. 96, 61–98 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  19. Lin, T., Lin, Y., Sun, W.: Error estimation of a class of quadratic immersed finite element methods for elliptic interface problems. Discret. Contin. Dynam. Syst. Ser. B 7, 807–823 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  20. Nevanlinna, O., Odeh, F.: Multiplier techniques for linear multistep methods. Numer. Funct. Anal. Optim. 3, 377–423 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  21. Sinha, R.K., Deka, B.: Optimal error estimates for linear parabolic problems with discontinuous coefficients. SIAM J. Numer. Anal. 43, 733–749 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  22. Sinha, R.K., Deka, B.: Finite element methods for semilinear elliptic and parabolic interface problems. Appl. Numer. Math. 59, 1870–1883 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  23. Wang, H., Liang, D., Ewing, R.E., Lyons, S.L., Qin, Guan: An approximation to miscible fluid flows in porous media with point sources and sinks by an Eulerian-Lagrangian localized adjoint method and mixed finite element methods. SIAM J. Sci. Comput. 22, 561–581 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Yang, C.: Convergence of a linearized second-order BDF-FEM for nonlinear parabolic interface problems. Comput. Math. Appl. 70, 265–281 (2015)

    Article  MathSciNet  Google Scholar 

  25. Zhang, Z., Yu, X.: Local discontinuous Galerkin method for parabolic interface problems. Acta Math. Appl. Sin. Eng. Ser. 31, 453–466 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  26. Ženíšek, A.: The finite element method for nonlinear elliptic equations with discontinuous coefficients. Numer. Math. 58, 51–77 (1990)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

I would like to thank the anonymous referees for the valuable comments and suggestions.

Author information

Authors and Affiliations

  1. Department of Mathematics, College of Science, Nanjing University of Posts and Telecommunications, Nanjing, 210023, Jiangsu, People’s Republic of China

    Chaoxia Yang

Authors
  1. Chaoxia Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chaoxia Yang.

Additional information

Communicated by Christian Lubich.

This work is supported in part by NSFC (Grant No.11401587) and NUPTSF (Grant No.NY214192). The research of the author was partially carried out at Universität Tübingen, supported by the Jiangsu Overseas Research & Training Program for University Prominent Young & Middle-aged Teachers and Presidents.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yang, C. Linearly implicit BDF methods for nonlinear parabolic interface problems. Bit Numer Math 57, 587–606 (2017). https://doi.org/10.1007/s10543-016-0641-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10543-016-0641-x

Keywords

Mathematics Subject Classification

Search

Navigation