Journal of Scientific Computing

, Volume 65, Issue 3, pp 1166–1188 | Cite as

Alternating Direction Implicit Galerkin Methods for an Evolution Equation with a Positive-Type Memory Term

  • Morrakot Khebchareon
  • Amiya K. Pani
  • Graeme Fairweather


We formulate and analyze new methods for the solution of a partial integrodifferential equation with a positive-type memory term. These methods combine the finite element Galerkin (FEG) method for the spatial discretization with alternating direction implicit (ADI) methods based on the Crank–Nicolson (CN) method and the second order backward differentiation formula for the time stepping. The ADI FEG methods are proved to be of optimal accuracy in time and in the \(L^2\) norm in space. Furthermore, the analysis is extended to include an ADI CN FEG method with a graded mesh in time for problems with a nonsmooth kernel. Numerical results confirm the predicted convergence rates and also exhibit optimal spatial accuracy in the \(L^{\infty }\) norm.


Partial integrodifferential equation Positive-type memory term Finite element Galerkin method Alternating direction implicit methods Optimal error estimates Smooth and nonsmooth kernels 

Mathematics Subject Classification

65M60 65M12 65M15 



The authors AKP and GF gratefully acknowledge the research support of the Department of Science and Technology, Government of India, through the National Programme on Differential Equations: Theory, Computation and Applications, DST Project No.SERB/F/1279/2011-2012. Support was also received by AKP from Chiangmai University, Thailand, and by GF from IIT Bombay while a Distinguished Visiting Professor at that institution.


  1. 1.
    Bramble, J.H., Ewing, R.E., Li, G.: Alternating direction multistep methods for parabolic problems-iterative stabilization. SIAM J. Numer. Anal. 25, 904–919 (1989)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Chen, C., Shih, T.: Finite Element Methods for Integrodifferential Equations. World Scientific, Singapore (1998)zbMATHCrossRefGoogle Scholar
  3. 3.
    Dendy Jr, J.E.: An analysis of some Galerkin schemes for the solution of nonlinear time-dependent problems. SIAM J. Numer. Anal. 12, 541–565 (1975)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Fernandes, R.I., Bialecki, B., Fairweather, G.: Alternating direction implicit orthogonal spline collocation methods for evolution equations. In: Jacob, M.J., Panda, S. (eds.) Mathematical Modelling and Applications to Industrial Problems (MMIP-2011), pp. 3–11. Macmillan Publishers India Limited, India (2012)Google Scholar
  5. 5.
    McLean, W., Mustapha, K.: A second order accurate numerical method for a fractional wave equation. Numer. Math. 105, 481–510 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    McLean, W., Thomée, V.: Numerical solution of an evolution equation with a positive-type memory term. J. Aust. Math. Soc. Ser. B. 35, 23–70 (1993)zbMATHCrossRefGoogle Scholar
  7. 7.
    McLean, W., Thomée, V.: Numerical solution via Laplace transforms of a fractional-order evolution equation. J. Integral Equ. Appl. 22, 57–94 (2010)zbMATHCrossRefGoogle Scholar
  8. 8.
    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. 30, 208–230 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    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)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Mustapha, K., McLean, W.: Discontinuous Galerkin method for an evolution equation with a memory term of positive type. Math. Comput. 78, 1975–1995 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Pani, A.K., Fairweather, G., Fernandes, R.I.: Alternating direction implicit orthogonal spline collocation methods for an evolution equation with a positive-type memory term. SIAM J. Numer. Anal. 46, 344–364 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Thomée, V.: A high order parallel method for time discretization of parabolic type equations based on Laplace transformation and quadrature. Int. J. Numer. Anal. Model. 2, 85–96 (2005)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Wheeler, M.F.: A priori \(L_2\) error estimates for Galerkin approximations to parabolic partial differential equations. SIAM J. Numer. Anal. 10, 723–759 (1973)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Morrakot Khebchareon
    • 1
  • Amiya K. Pani
    • 2
  • Graeme Fairweather
    • 3
  1. 1.Department of Mathematics, Faculty of ScienceChiang Mai UniversityChiang MaiThailand
  2. 2.Department of Mathematics, Industrial Mathematics GroupIIT BombayPowai, MumbaiIndia
  3. 3.Mathematical Reviews, American Mathematical SocietyAnn ArborUSA

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