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Journal of Scientific Computing

, Volume 79, Issue 1, pp 414–441 | Cite as

A Posteriori Error Analysis of the Crank–Nicolson Finite Element Method for Parabolic Integro-Differential Equations

  • G. Murali Mohan ReddyEmail author
  • Rajen Kumar Sinha
  • José Alberto Cuminato
Article
  • 76 Downloads

Abstract

We study a posteriori error analysis for the space-time discretizations of linear parabolic integro-differential equation in a bounded convex polygonal or polyhedral domain. The piecewise linear finite element spaces are used for the space discretization, whereas the time discretization is based on the Crank–Nicolson method. The Ritz–Volterra reconstruction operator (IMA J Numer Anal 35:341–371, 2015), a generalization of elliptic reconstruction operator (SIAM J Numer Anal 41:1585–1594, 2003), is used in a crucial way to obtain optimal rate of convergence in space. Moreover, a quadratic (in time) space-time reconstruction operator is introduced to establish second order convergence in time. The proposed method uses nested finite element spaces and the standard energy technique to obtain optimal order error estimator in the \(L^{\infty }(L^2)\)-norm. Numerical experiments are performed to validate the optimality of the error estimators.

Keywords

Parabolic integro-differential equations Finite element method Ritz–Volterra reconstruction Crank–Nicolson method A posteriori error estimate 

Notes

Acknowledgements

The authors wish to thank both the referees for their valuable comments and suggestion which led to the improvement of this manuscript. G. Murali Mohan Reddy would like to thank FAPESP for the financial support received (Grant No. 2016/19648-9).

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • G. Murali Mohan Reddy
    • 1
    Email author
  • Rajen Kumar Sinha
    • 2
  • José Alberto Cuminato
    • 1
  1. 1.Department of Applied Mathematics and Statistics, Institute of Mathematics and Computer SciencesUniversity of São Paulo at São CarlosSão CarlosBrazil
  2. 2.Department of MathematicsIndian Institute of Technology GuwahatiGuwahatiIndia

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