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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
Article

Abstract

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.

Keywords

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 

Notes

Acknowledgments

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.

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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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