Skip to main content
SpringerLink
Log in

On the stability of the ale space-time discontinuous Galerkin method for nonlinear convection-diffusion problems in time-dependent domains

  • Published:
Applications of Mathematics Aims and scope Submit manuscript

Abstract

The paper is concerned with the analysis of the space-time discontinuous Galerkin method (STDGM) applied to the numerical solution of the nonstationary nonlinear convection-diffusion initial-boundary value problem in a time-dependent domain formulated with the aid of the arbitrary Lagrangian-Eulerian (ALE) method. In the formulation of the numerical scheme we use the nonsymmetric, symmetric and incomplete versions of the space discretization of diffusion terms and interior and boundary penalty. The nonlinear convection terms are discretized with the aid of a numerical flux. The space discretization uses piecewise polynomial approximations of degree not greater than p with an integer p ⩾ 1. In the theoretical analysis, the piecewise linear time discretization is used. The main attention is paid to the investigation of unconditional stability of the method.

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. G. Akrivis, C. Makridakis: Galerkin time-stepping methods for nonlinear parabolic equations. M2AN, Math. Model. Numer. Anal. 38 (2004), 261–289.

    Article  MathSciNet  MATH  Google Scholar 

  2. D. N. Arnold, F. Brezzi, B. Cockburn, L. D. Marini: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39 (2002), 1749–1779.

    Article  MathSciNet  MATH  Google Scholar 

  3. I. Babuška, C. E. Baumann, J. T. Oden: A discontinuous hp finite element method for diffusion problems: 1-D analysis. Comput. Math. Appl. 37 (1999), 103–122.

    Article  MathSciNet  MATH  Google Scholar 

  4. M. Balazsova, M. Feistauer, M. Hadrava, A. Kosik: On the stability of the space-time discontinuous Galerkin method for the numerical solution of nonstationary nonlinear convection-diffusion problems. To appear in J. Numer. Math.

  5. F. Bassi, S. Rebay: A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations. J. Comput. Phys. 131 (1997), 267–279.

    Article  MathSciNet  MATH  Google Scholar 

  6. C. E. Baumann, J. T. Oden: A discontinuous hp finite element method for the Euler and Navier-Stokes equations. Int. J. Numer. Methods Fluids 31 (1999), 79–95.

    Article  MathSciNet  MATH  Google Scholar 

  7. D. Boffi, L. Gastaldi, L. Heltai: Numerical stability of the finite element immersed boundary method. Math. Models Methods Appl. Sci. 17 (2007), 1479–1505.

    Article  MathSciNet  MATH  Google Scholar 

  8. A. Bonito, I. Kyza, R. H. Nochetto: Time-discrete higher-order ALE formulations: stability. SIAM J. Numer. Anal. 51 (2013), 577–604.

    Article  MathSciNet  MATH  Google Scholar 

  9. F. Brezzi, G. Manzini, D. Marini, P. Pietra, A. Russo: Discontinuous Galerkin approximations for elliptic problems. Numer. Methods Partial Differ. Equations 16 (2000), 365–378.

    Article  MathSciNet  MATH  Google Scholar 

  10. J. Česenek, M. Feistauer: Theory of the space-time discontinuous Galerkin method for nonstationary parabolic problems with nonlinear convection and diffusion. SIAM J. Numer. Anal. 50 (2012), 1181–1206.

    Article  MathSciNet  MATH  Google Scholar 

  11. J. Česenek, M. Feistauer, J. Horaček, V. Kučera, J. Prokopova: Simulation of compressible viscous flow in time-dependent domains. Appl. Math. Comput. 219 (2013), 7139–7150.

    Article  MathSciNet  MATH  Google Scholar 

  12. J. Česenek, M. Feistauer, A. Kosik: DGFEM for the analysis of airfoil vibrations induced by compressible flow. ZAMM, Z. Angew. Math. Mech. 93 (2013), 387–402.

    Article  MathSciNet  MATH  Google Scholar 

  13. K. Chrysafinos, N. J. Walkington: Error estimates for the discontinuous Galerkin methods for parabolic equations. SIAM J. Numer. Anal. 44 (2006), 349–366.

    Article  MathSciNet  MATH  Google Scholar 

  14. B. Cockburn, C.-W. Shu: Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput. 16 (2001), 173–261.

    Article  MathSciNet  MATH  Google Scholar 

  15. V. Dolejši: On the discontinuous Galerkin method for the numerical solution of the Navier-Stokes equations. Int. J. Numer. Methods Fluids 45 (2004), 1083–1106.

    Article  MATH  Google Scholar 

  16. V. Dolejši, M. Feistauer: Discontinuous Galerkin Method-Analysis and Applications to Compressible Flow. Springer, Heidelberg, 2015.

    Book  Google Scholar 

  17. V. Dolejši, M. Feistauer, J. Hozman: Analysis of semi-implicit DGFEM for nonlinear convection-diffusion problems on nonconforming meshes. Comput. Methods Appl. Mech. Eng. 196 (2007), 2813–2827.

    Article  MATH  Google Scholar 

  18. J. Donea, S. Giuliani, J. P. Halleux: An arbitrary Lagrangian-Eulerian finite element method for transient dynamic fluid-structure interactions. Comput. Methods Appl. Mech. Eng. 33 (1982), 689–723.

    Article  MATH  Google Scholar 

  19. K. Eriksson, D. Estep, P. Hansbo, C. Johnson: Computational differential equations. Cambridge Univ. Press, Cambridge, 1996.

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  21. D. Estep, S. Larsson: The discontinuous Galerkin method for semilinear parabolic problems. RAIRO, Modélisation Math. Anal. Numér. 27 (1993), 35–54.

    MathSciNet  MATH  Google Scholar 

  22. M. Feistauer, J. Felcman, I. Straškraba: Mathematical and Computational Methods for Compressible Flow. Numerical Mathematics and Scientific Computation, Oxford University Press, Oxford, 2003.

    Google Scholar 

  23. M. Feistauer, J. Hajek, K. Švadlenka: Space-time discontinuous Galerkin method for solving nonstationary convection-diffusion-reaction problems. Appl. Math., Praha 52 (2007), 197–233.

    Article  MATH  Google Scholar 

  24. M. Feistauer, J. Hasnedlova-Prokopova, J. Horaček, A. Kosik, V. Kučera: DGFEM for dynamical systems describing interaction of compressible fluid and structures. J. Comput. Appl. Math. 254 (2013), 17–30.

    Article  MathSciNet  MATH  Google Scholar 

  25. M. Feistauer, J. Horaček, V. Kučera, J. Prokopova: On numerical solution of compressible flow in time-dependent domains. Math. Bohem. 137 (2012), 1–16.

    MathSciNet  MATH  Google Scholar 

  26. M. Feistauer, V. Kučera, K. Najzar, J. Prokopova: Analysis of space-time discontinuous Galerkin method for nonlinear convection-diffusion problems. Numer. Math. 117 (2011), 251–288.

    Article  MathSciNet  MATH  Google Scholar 

  27. M. Feistauer, V. Kučera, J. Prokopova: Discontinuous Galerkin solution of compressible flow in time-dependent domains. Math. Comput. Simul. 80 (2010), 1612–1623.

    Article  MATH  Google Scholar 

  28. L. Formaggia, F. Nobile: A stability analysis for the arbitrary Lagrangian Eulerian formulation with finite elements. East-West J. Numer. Math. 7 (1999), 105–131.

    MathSciNet  MATH  Google Scholar 

  29. L. Gastaldi: A priori error estimates for the arbitrary Lagrangian Eulerian formulation with finite elements. East-West J. Numer. Math. 9 (2001), 123–156.

    MathSciNet  MATH  Google Scholar 

  30. J. Hasnedlova, M. Feistauer, J. Horaček, A. Kosik, V. Kučera: Numerical simulation of fluid-structure interaction of compressible flow and elastic structure. Computing 95 (2013), S343–S361.

    Article  Google Scholar 

  31. O. Havle, V. Dolejši, M. Feistauer: Discontinuous Galerkin method for nonlinear convection-diffusion problems with mixed Dirichlet-Neumann boundary conditions. Appl. Math., Praha 55 (2010), 353–372.

    Article  MATH  Google Scholar 

  32. P. Houston, C. Schwab, E. Suli: Discontinuous hp-finite element methods for advection-diffusion-reaction problems. SIAM J. Numer. Anal. 39 (2002), 2133–2163.

    Article  MathSciNet  MATH  Google Scholar 

  33. K. Khadra, P. Angot, S. Parneix, J.-P. Caltagirone: Fictiuous domain approach for numerical modelling of Navier-Stokes equations. Int. J. Numer. Methods Fluids 34 (2000), 651–684.

    Article  MATH  Google Scholar 

  34. J. T. Oden, I. Babuška, C. E. Baumann: A discontinuous hp finite element method for diffusion problems. J. Comput. Phys. 146 (1998), 491–519.

    Article  MathSciNet  MATH  Google Scholar 

  35. D. Schotzau: hp-DGFEM for Parabolic Evolution Problems. Applications to Diffusion and Viscous Incompressible Fluid Flow. PhD Thesis, ETH No. 13041, Zürich, 1999.

  36. D. Schotzau, C. Schwab: An hp a priori error analysis of the DG time-stepping method for initial value problems. Calcolo 37 (2000), 207–232.

    Article  MathSciNet  Google Scholar 

  37. V. Thomee: Galerkin Finite Element Methods for Parabolic Problems. Springer Series in Computational Mathematics 25, Springer, Berlin, 2006.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Mathematics and Physics, Charles University in Prague, Sokolovská 83, 186 75, Praha 8, Czech Republic

    Monika Balázsová & Miloslav Feistauer

Authors
  1. Monika Balázsová
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Miloslav Feistauer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Monika Balázsová.

Additional information

The research of M. Feistauer was supported by the grant 13-00522S of the Czech Science Foundation and the research of M.Balázsová was supported by the Charles University in Prague, project GA UK No. 127615.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Balázsová, M., Feistauer, M. On the stability of the ale space-time discontinuous Galerkin method for nonlinear convection-diffusion problems in time-dependent domains. Appl Math 60, 501–526 (2015). https://doi.org/10.1007/s10492-015-0109-3

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10492-015-0109-3

Keywords

MSC 2010

Search

Navigation