Skip to main content
SpringerLink
Log in

An Optimal Order CG-DG Space-Time Discretization Method for Parabolic Problems

  • Chapter
  • First Online:
Advanced Finite Element Methods with Applications (FEM 2017)

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 128))

Included in the following conference series:

  • 975 Accesses

Abstract

We consider a space-time discretization method for second-order parabolic problems with inhomogeneous (time-dependent) Dirichlet boundary conditions. A combination of a temporal discontinuous Galerkin scheme and a spatial continuous Galerkin scheme is used. In previous work it has been established that the standard semi-discrete temporal scheme has to be modified to obtain an optimal error bound. Here we extend this modification to a fully discrete scheme. For this modified discretization an optimal error bound for the energy norm is derived. Results of experiments confirm the theoretically predicted optimal convergence rates. We are able to pinpoint why the standard CG-DG space-time method (without any modifications) has suboptimal convergence behavior. The method presented here avoids this suboptimality in a way which is computationally very cheap.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 99.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 129.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 129.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Alonso-Mallo, I., Cano, B.: Spectral/Rosenbrock discretizations without order reduction for linear parabolic problems. Appl. Numer. Math. 41(2), 247–268 (2002)

    Article  MathSciNet  Google Scholar 

  2. Altmann, R., Zimmer, C.: Runge-Kutta methods for linear semi-explicit operator differential-algebraic equations. Math. Comput. 87(309), 149–174 (2017)

    Article  MathSciNet  Google Scholar 

  3. Boffi, D., Brezzi, F., Fortin, M.: Mixed Finite Element Methods and Applications. Springer, Berlin (2013)

    Book  Google Scholar 

  4. Di Pietro, D.A., Ern, A.: Mathematical Aspects of Discontinuous Galerkin Methods. Springer, Berlin (2012)

    Book  Google Scholar 

  5. Eriksson, K., Johnson, C.: Adaptive finite element methods for parabolic problems I: a linear model problem. SIAM J. Numer. Anal. 28(1), 43–77 (1991)

    Article  MathSciNet  Google Scholar 

  6. Eriksson, K., Johnson, C.: Adaptive finite element methods for parabolic problems II: optimal error estimates in L ∞ L 2 and L ∞ L ∞. SIAM J. Numer. Anal. 32(3), 706–740 (1995)

    Article  MathSciNet  Google Scholar 

  7. Eriksson, K., Johnson, C., Larsson, S.: Adaptive finite element methods for parabolic problems VI: analytic semigroups. SIAM J. Numer. Anal. 35(4), 1315–1325 (1998)

    Article  MathSciNet  Google Scholar 

  8. Ern, A., Guermond, J.L.: Theory and Practice Of Finite Elements. Applied Mathematical Sciences, vol. 159. Springer, New York (2004)

    Google Scholar 

  9. Evans, L.C.: Partial Differential Equations. Graduate Studies in Mathematics, vol. 19. American Mathematical Society, Providence (2010)

    Google Scholar 

  10. Groß, S., Peters, J., Reichelt, V., Reusken, A.: The DROPS package for numerical simulations of incompressible flows using parallel adaptive multigrid techniques. Preprint 227, IGPM, RWTH Aachen University (2002)

    Google Scholar 

  11. Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods. Springer, New York (2008)

    Book  Google Scholar 

  12. Jamet, P.: Galerkin-type approximations which are discontinuous in time for parabolic equations in a variable domain. SIAM J. Numer. Anal. 15(5), 912–928 (1978)

    Article  MathSciNet  Google Scholar 

  13. Kennedy, C.A., Carpenter, M.H.: Diagonally implicit Runge-Kutta methods for ordinary differential equations. A review. Tech. Rep. NASA/TM–2016–219173, NASA (2016)

    Google Scholar 

  14. Larsson, S., Thomée, V., Wahlbin, L.B.: Numerical solution of parabolic integro-differential equations by the discontinuous Galerkin method. Math. Comput. 67(221), 45–71 (1998)

    Article  MathSciNet  Google Scholar 

  15. Schötzau, D., Schwab, C.: Time discretization of parabolic problems by the hp-version of the discontinuous Galerkin finite element method. SIAM J. Numer. Anal. 38(3), 837–875 (2000)

    Article  MathSciNet  Google Scholar 

  16. Thomée, V.: Galerkin Finite Element Methods for Parabolic Problems. Springer Series in Computational Mathematics, vol. 25. Springer, Berlin (2006)

    Google Scholar 

  17. Voulis, I., Reusken, A.: Discontinuous Galerkin time discretization methods for parabolic problems with linear constraints. J. Numer. Math. (2018). https://doi.org/10.1515/jnma-2018-0013

  18. Wloka, J.: Partial Differential Equations. Cambridge University Press, Cambridge (1987)

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. RWTH-Aachen University, Institut fuer Geometrie und Praktische Mathematik, Aachen, Germany

    Igor Voulis

Authors
  1. Igor Voulis
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Igor Voulis .

Editor information

Editors and Affiliations

  1. Institut für Mathematik & Computergestützte Simulation, Universität der Bundeswehr München, Neubiberg, Germany

    Thomas Apel

  2. Institute for Computational Mathematics, Johannes Kepler University Linz, Linz, Austria

    Ulrich Langer

  3. Fakultät für Mathematik, TU Chemnitz, Chemnitz, Germany

    Arnd Meyer

  4. Institut für Angewandte Mathematik, Technische Universität Graz, Graz, Austria

    Olaf Steinbach

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Voulis, I. (2019). An Optimal Order CG-DG Space-Time Discretization Method for Parabolic Problems. In: Apel, T., Langer, U., Meyer, A., Steinbach, O. (eds) Advanced Finite Element Methods with Applications. FEM 2017. Lecture Notes in Computational Science and Engineering, vol 128. Springer, Cham. https://doi.org/10.1007/978-3-030-14244-5_18

Download citation

Publish with us

Policies and ethics

Search

Navigation