Skip to main content
SpringerLink
Log in

Optimal Error Estimates of Two Mixed Finite Element Methods for Parabolic Integro-Differential Equations with Nonsmooth Initial Data

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

Abstract

In the first part of this article, a new mixed method is proposed and analyzed for parabolic integro-differential equations (PIDE) with nonsmooth initial data. Compared to the standard mixed method for PIDE, the present method does not bank on a reformulation using a resolvent operator. Based on energy arguments combined with a repeated use of an integral operator and without using parabolic type duality technique, optimal \(L^2\)-error estimates are derived for semidiscrete approximations, when the initial condition is in \(L^2\). Due to the presence of the integral term, it is, further, observed that a negative norm estimate plays a crucial role in our error analysis. Moreover, the proposed analysis follows the spirit of the proof techniques used in deriving optimal error estimates for finite element approximations to PIDE with smooth data and therefore, it unifies both the theories, i.e., one for smooth data and other for nonsmooth data. Finally, we extend the proposed analysis to the standard mixed method for PIDE with rough initial data and provide an optimal error estimate in \(L^2,\) which improves upon the results available in the literature.

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

Similar content being viewed by others

References

  1. 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  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  3. Cannon, J.R., Lin, Y.: Nonlocal \(H^1\) projections and Galerkin methods for nonlinear parabolic integro-differential equations. Calcolo 25, 187–201 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chen, Z.: Expanded mixed finite element methods for linear second order elliptic problems-I, RAIRO- M2AN 32, pp. 479–499 (1998)

  5. Chen, Z.: Expanded mixed finite element methods for quasilinear second order elliptic problems-II, RAIRO- M2AN 32, pp. 501–520 (1998)

  6. Cushman, J.H., Glinn, T.R.: Nonlocal dispersion in media with continuously evolving scales of heterogeneity. Transp. Porous Media 13, 123–138 (1993)

    Article  Google Scholar 

  7. Dagan, G.: The significance of heterogeneity of evolving scales to transport in porous formations. Water Resour. Res. 30, 3327–3336 (1994)

    Article  Google Scholar 

  8. Ewing, R.E., Lazarov, R.D., Lin, Y.: Finite volume element approximations of integro-differential parabolic problems, Recent advances in numerical methods and applications, II (Sofia, 1998), 3–15. World Sci. Publ., River Edge (1999)

    Google Scholar 

  9. Ewing, R.E., Lazarov, R., Lin, Y.: Finite volume element approximations of nonlocal reactive flows in porous media. Numer. Meth. PDE. 16, 285–311 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ewing, R.E., Lazarov, R., Lin, Y.: Finite volume element approximations of nonlocal in time one-dimensional flows in porous media. Computing 64, 157–182 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ewing, R.E., Lin, Y., Sun, T., Wang, J., Zhang, S.: Sharp \(L^2\)-error estimates and superconvergence of mixed finite element methods for non-Fickian flows in porous media. SIAM J. Numer. Anal. 40, 1538–1560 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ewing, R.E., Lin, Y., Wang, J.: A numerical approximation of non-Fickian flows with mixing length growth in porous media. Acta. Math. Univ. Comenian. 70, 75–84 (2001)

    MathSciNet  MATH  Google Scholar 

  13. Ewing, R.E., Lin, Y., Wang, J., Zhang, S.: \(L^\infty \)-error estimates and superconvergence in maximum norm of mixed finite element methods for non-Fickian flows in porous media. Int. J. Numer. Anal. Model. 2, 301–328 (2005)

    MathSciNet  MATH  Google Scholar 

  14. Goswami, D., Pani, A.K.: An alternate approach to optimal \(L^2\)-error analysis of semidiscrete Galerkin methods for linear parabolic problems with nonsmooth initial data. Numer. Funct. Anal. Optimz. 32, 946–982 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Jiang, Z.: \(L^{\infty }(L^2)\) and \(L^{\infty }(L^{\infty })\) error estimates for mixed methods for integro-differential equations of parabolic type, M2AN Math. Model. Numer. Anal. 33, 531–546 (1999)

    Article  MATH  Google Scholar 

  16. Lin, Y., Thomée, V., Wahlbin, L.B.: Ritz-Volterra projections to finite element spaces and applications to integro-differential and related equations. SIAM J. Numer. Anal. 28, 1040–1070 (1991)

    Article  Google Scholar 

  17. Pani, A.K., Thomée, V., Wahlbin, L.B.: Numerical methods for hyperbolic and parabolic integro-differential equations. J. Integr. Equ. Appl. 4, 533–584 (1992)

    Article  MATH  Google Scholar 

  18. Pani, A.K., Sinha, R.K.: Error estimate for semidiscrete Galerkin approximation to a time dependent parabolic integro-differential equation with nonsmooth data. Calcolo 37, 181–205 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  19. Pani, A.K., Peterson, T.E.: Finite element methods with numerical quadrature for parabolic integro- differential equations. SIAM J. Numer. Anal. 33, 1084–1105 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  20. Renardy, M., Hrusa, W. and Nohel, J., Mathematical problems in viscoelasticity, Pitman Monographs and Survey in Pure Appl. Math. Vol. 35, New York, Wiley (1987)

  21. Sinha, R. K., Ewing, R. E. and Lazarov, R. D. Some new error estimates of a semidiscrete finite volume element method for a parabolic integro-differential equation with nonsmooth initial data, SIAM J. Numer. Anal. 43, 2320–2343 (electronic) (2006)

    Google Scholar 

  22. Sinha, R.K., Ewing, R.E., Lazarov, R.D.: Mixed finite element approximations of parabolic integro-differential equations with nonsmooth initial data. SIAM J. Numer. Anal. 47, 3269–3292 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  23. Thomée, V., Zhang, N.Y.: Error estimates for semidiscrete finite element methods for parabolic integro-differential equations. Math. Comp. 53, 121–139 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  24. Woodward, C.S., Dawson, C.N.: Analysis of expanded finite element methods for a nonlinear parabolic equation modelling flow into variably saturated porous media. SIAM J. Numer. Anal. 37, 701–724 (2000)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The first author would like to thank CSIR, Government of India for the financial support. The second author acknowledges the research support of the Department of Science and Technology, Government of India under DST-CNPq Indo-Brazil Project No. DST/INT/Brazil /RPO-05/2007 ( Grant No. 490795/2007-2). This publication is also based on the work (AKP) supported in part by Award No. KUK-C1-013-04, made by King Abdullah University of Science and Technology (KAUST). The authors also thank the referees for their valuable suggestions.

Author information

Authors and Affiliations

  1. Departamento de Matemática, UFPR, Centro Politécnico, Jardim das Américas, Caixa Postal 19081, CEP: 81531-980,  Curitiba, Paran, Brazil

    Deepjyoti Goswami

  2. Department of Mathematics, Industrial Mathematics Group, Indian Institute of Technology Bombay, Powai, Mumbai, 400076, India

    Amiya K. Pani

  3. Department of Mathematics, Birla Institute of Technology and Science, Pilani Pilani,  333031, Rajasthan, India

    Sangita Yadav

Authors
  1. Deepjyoti Goswami
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Amiya K. Pani
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sangita Yadav
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Amiya K. Pani.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Goswami, D., Pani, A.K. & Yadav, S. Optimal Error Estimates of Two Mixed Finite Element Methods for Parabolic Integro-Differential Equations with Nonsmooth Initial Data. J Sci Comput 56, 131–164 (2013). https://doi.org/10.1007/s10915-012-9666-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-012-9666-8

Keywords

Search

Navigation