Numerische Mathematik

, Volume 124, Issue 1, pp 151–182 | Cite as

Efficient numerical realization of discontinuous Galerkin methods for temporal discretization of parabolic problems

Article

Abstract

We present an efficient and easy to implement approach to solving the semidiscrete equation systems resulting from time discretization of nonlinear parabolic problems with discontinuous Galerkin methods of order \(r\). It is based on applying Newton’s method and decoupling the Newton update equation, which consists of a coupled system of \(r+1\) elliptic problems. In order to avoid complex coefficients which arise inevitably in the equations obtained by a direct decoupling, we decouple not the exact Newton update equation but a suitable approximation. The resulting solution scheme is shown to possess fast linear convergence and consists of several steps with same structure as implicit Euler steps. We construct concrete realizations for order one to three and give numerical evidence that the required computing time is reduced significantly compared to assembling and solving the complete coupled system by Newton’s method.

Mathematics Subject Classification (2000)

65M99 65M60 65F10 

Notes

Acknowledgments

Andreas Springer gratefully acknowledges financial support from the Munich Centre of Advanced Computing and the International Graduate School of Science and Engineering at the Technische Universität München.

References

  1. 1.
    Becker, R., Meidner, D., Vexler, B.: Efficient numerical solution of parabolic optimization problems by finite element methods. Optim. Methods Softw. 22(5), 813–833 (2007)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Besier, M., Rannacher, R.: Goal-oriented space-time adaptivity in the finite element Galerkin method for the computation of nonstationary incompressible flow. Int. J. Numer. Methods Fluids 458, 2735–2757 (2012)MathSciNetGoogle Scholar
  3. 3.
    Brezinski, C., Van Iseghem, J.: A taste of Padé approximation. In: Acta numerica, 1995, Acta Numer., pp. 53–103. Cambridge University Press, Cambridge (1995)Google Scholar
  4. 4.
    Chrysafinos, K.: Discontinuous Galerkin approximations for distributed optimal control problems constrained by parabolic PDEs. Int. J. Numer. Anal. Model. 4(3–4), 690–712 (2007)MathSciNetMATHGoogle Scholar
  5. 5.
    Ciarlet, P.G.: The finite element method for elliptic problems. Classics in Applied Mathematics, vol. 40. SIAM, Philadelphia, PA (2002)Google Scholar
  6. 6.
    Dunford, N., Schwartz, J.T.: Linear Operators: Part I. Wiley Classics Library. Wiley, New York (1988)Google Scholar
  7. 7.
    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)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Eriksson, K., Johnson, C.: Adaptive finite element methods for parabolic problems II: Optimal error estimates in \(L_\infty L_2\) and \(L_\infty L_\infty \). SIAM J. Numer. Anal. 32(3), 706–740 (1995)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Eriksson, K., Johnson, C., Thomée, V.: Time discretization of parabolic problems by the discontinuous Galerkin method. M2AN Math. Model. Numer. Anal. 19, 611–643 (1985)MATHGoogle Scholar
  10. 10.
    Frank, R., Schneid, J., Überhuber, C.W.: Order results for implicit Runge–Kutta methods applied to stiff systems. SIAM J. Numer. Anal. 22(3), 515–534 (1985)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Hairer, E., Wanner, G.: Stiff differential equations solved by Radau methods. J. Comput. Appl. Math. 111(1–2), 93–111 (1999)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Hussain, S., Schieweck, F., Turek, S.: Higher order Galerkin time discretizations and fast multigrid solvers for the heat equation. J. Numer. Math. 19(1), 41–62 (2011)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Lang, J.: Adaptive Multilevel Solution of Nonlinear Parabolic PDE Systems. Lecture Notes in Computational Science and Engineering, vol. 16. Springer, Berlin (2001)Google Scholar
  14. 14.
    Lesaint, P., Raviart, P.-A.: On a finite element method for solving the neutron transport equation. In: Mathematical Aspects of Finite Elements in Partial Differential Equations, pp. 89–123. Academic Press, New York (1974)Google Scholar
  15. 15.
    Meidner, D., Vexler, B.: Adaptive space–time finite element methods for parabolic optimization problems. SIAM J. Control Optim. 46(1), 116–142 (2007)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Meidner, D., Vexler, B.: A priori error estimates for space–time finite element approximation of parabolic optimal control problems. Part I: problems without control constraints. SIAM J. Control Optim. 47(3), 1150–1177 (2008)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Meidner, D., Vexler, B.: A priori error estimates for space–time finite element approximation of parabolic optimal control problems. Part II: problems with control constraints. SIAM J. Control Optim. 47(3), 1301–1329 (2008)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Meidner, D., Vexler, B.: A priori error analysis of the Petrov–Galerkin Crank–Nicolson Scheme for parabolic optimal control problems. SIAM J. Control Optim. 49(5), 2183–2211 (2011)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Neitzel, I., Vexler, B.: A priori error estimates for space-time finite element discretization of semilinear parabolic optimal control problems. Numer. Math. 120(2), 345–386 (2011)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Perron, O.: Die Lehre von den Kettenbrüchen. Band II. Analytisch-funktionentheoretische Kettenbrüche, 3rd edn. B. G. Teubner Verlagsgesellschaft, Stuttgart (1957)Google Scholar
  21. 21.
    Raymond, J.P., Zidani, H.: Hamiltonian Pontryagin’s principles for control problems governed by semilinear parabolic equations. Appl. Math. Optim. 39(2), 143–177 (1999)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Schmich, M., Vexler, B.: Adaptivity with dynamic meshes for space–time finite element discretizations of parabolic equations. SIAM J. Sci. Comput. 30(1), 369–393 (2008)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    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)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Schötzau, D., Wihler, T.: A posteriori error estimation for hp-version time-stepping methods for parabolic partial differential equations. Numer. Math. 115, 475–509 (2010)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Gascoigne: The finite element toolkit. http://www.gascoigne.uni-hd.de
  26. 26.
    RoDoBo: A C++ library for optimization with stationary and nonstationary PDEs with interface to Gascoigne [25]. http://www.rodobo.uni-hd.de
  27. 27.
    Thomée, V.: Galerkin Finite Element Methods for Parabolic Problems. Spinger Series in Computational Mathematics, vol. 25. Springer, Berlin (2006)Google Scholar
  28. 28.
    Wanner, G., Hairer, E., Nørsett, S.P.: Order stars and stability theorems. BIT 18(4), 475–489 (1978)MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Wathen, A.J.: Realistic eigenvalue bounds for the Galerkin mass matrix. IMA J. Numer. Anal. 7(4), 449–457 (1987)MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Werder, T., Gerdes, K., Schötzau, D., Schwab, C.: \(hp\)-Discontinuous Galerkin time stepping for parabolic problems. Comput. Methods Appl. Mech. Engrg. 190(49–50), 6685–6708 (2001)MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Wloka, J.: Partial Differential Equations. Cambridge University Press, Cambridge (1987) (Translated from the German by C.B. Thomas and M.J. Thomas)Google Scholar
  32. 32.
    Ypma, T.J.: Local convergence of inexact Newton methods. SIAM J. Numer. Anal. 21(3), 583–590 (1984)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Thomas Richter
    • 1
  • Andreas Springer
    • 2
  • Boris Vexler
    • 2
  1. 1.Institut für angewandte Mathematik, Ruprecht-Karls-Universität HeidelbergHeidelbergGermany
  2. 2.Lehrstuhl für Mathematische Optimierung, Technische Universität München, Fakultät für MathematikGarching b. MünchenGermany

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