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Numerical Algorithms

, Volume 52, Issue 1, pp 69–88 | Cite as

Convergence analysis of a discontinuous Galerkin method for a sub-diffusion equation

  • William McLean
  • Kassem Mustapha
Original Paper

Abstract

We employ a piecewise-constant, discontinuous Galerkin method for the time discretization of a sub-diffusion equation. Denoting the maximum time step by k, we prove an a priori error bound of order k under realistic assumptions on the regularity of the solution. We also show that a spatial discretization using continuous, piecewise-linear finite elements leads to an additional error term of order h 2 max (1,logk  − 1). Some simple numerical examples illustrate this convergence behaviour in practice.

Keywords

Non-uniform time steps Memory term Finite elements 

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References

  1. 1.
    Cuesta, E., Lubich, C., Palencia, C.: Convolution quadrature time discretization of fractional diffusive-wave equations. Math. Comput. 75, 673–696 (2006)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Langlands, T.A.M., Henry, B.I.: The accuracy and stability of an implicit solution method for the fractional diffusion equation. J. Comp. Phys. 205, 719–936 (2005)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    López-Fernández, M., Lubich, C., Schädle, A.: Adaptive, fast and oblivious convolution quadrature in evolution equations with memory. SIAM J. Sci. Comput. 30, 1015–1037 (2008)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    López-Fernandez, M., Palencia, C., Schädle, A.: A spectral order method for inverting sectorial Laplace transforms. SIAM J. Numer. Anal. 44, 1332–1350 (2006)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1–77 (2000)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    McLean, W., Thomée, V.: Numerical solution via Laplace transforms of a fractional order evolution equation. J. Integral Equ. Appl. (2008, in press)Google Scholar
  7. 7.
    McLean, W., Thomée, V.: Maximum-norm error analysis of a numerical solution via Laplace transformation and quadrature of a fractional order evolution equation. IMA J. Numer. Anal. (2008, in press)Google Scholar
  8. 8.
    McLean, W., Thomée, V., Wahlbin, L.B.: Discretization with variable time steps of an evolution equation with a positive-type memory term. J. Comput. Appl. Math. 69, 49–69 (1996)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Mustapha, K., McLean, W.: Discontinuous Galerkin method for an evolution equation with a memory term of positive type. Math. Comput. (2008, in press)Google Scholar
  10. 10.
    Schädle, A., López-Fernandez, M., Lubich, C.: Fast and oblivious convolution quadrature. SIAM J. Sci. Comput. 28, 421–438 (2006)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2008

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsThe University of New South WalesSydneyAustralia
  2. 2.Department of Mathematics and StatisticsKing Fahd University of Petroleum and MineralsDhahranSaudi Arabia

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