Journal of Applied Mathematics and Computing

, Volume 53, Issue 1–2, pp 413–443 | Cite as

Fully discrete finite element scheme for nonlocal parabolic problem involving the Dirichlet energy

  • Vimal Srivastava
  • Sudhakar Chaudhary
  • V. V. K. Srinivas Kumar
  • Balaji Srinivasan
Original Research
First Online:
Received:

Abstract

In this article we present a finite element scheme for solving a nonlocal parabolic problem involving the Dirichlet energy. For time discretization, we use backward Euler method. The nonlocal term causes difficulty while using Newton’s method. Indeed, after applying Newton’s method we get a full Jacobian matrix due to the nonlocal term. In order to avoid this difficulty we use the technique given by Gudi (SIAM J Numer Anal 50(2):657–668, 2012) for elliptic nonlocal problem of Kirchhoff type. We discuss the well-posedness of the weak formulation at continuous as well as at discrete levels. We also derive a priori error estimates for both semi-discrete and fully discrete formulations. Results based on the usual finite element method are provided to confirm the theoretical estimates.

Keywords

Nonlocal Kirchhoff equation Backward Euler method Newton iteration method 

Mathematics Subject Classifications

65N12 65N15 65N22 35N30 
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Notes

Acknowledgments

The authors acknowledge anonymous reviewers for many helpful suggestions and comments. Also, the authors would like to sincerely thank Dr. T. Gudi for his valuable suggestions. The second author’s work is supported by National Board for Higher Mathematics, DAE,(Grant No. 2/40(26)/2014/R&D-II/9598) India.

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Copyright information

© Korean Society for Computational and Applied Mathematics 2015

Authors and Affiliations

  • Vimal Srivastava
    • 1
  • Sudhakar Chaudhary
    • 1
  • V. V. K. Srinivas Kumar
    • 1
  • Balaji Srinivasan
    • 2
  1. 1.Department of MathematicsIndian Institute of TechnologyDelhiIndia
  2. 2.Department of Applied MechanicsIndian Institute of TechnologyDelhiIndia

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