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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Chipot, M., Lavot, B.: Remarks on a nonlocal problem involving the Dirichlet energy. Rend. Sem. Mat. Univ. Padova 110, 199–220 (2003)
Zheng, S., Chipot, M.: Asymptotic behavior of solutions to nonlinear parabolic equations with nonlocal terms. Asymptot. Anal. 45, 301–312 (2005)
Robalo, R.J., Almeida, R.M.P., do Carmo Coimbra, M., Ferreira, J.: A reaction diffusion model for a class of nonlinear parabolic equations with moving boundaries: existence, uniqueness, exponential decay and simulation. Appl. Math. Model. 38(23), 5609–5622 (2014)
Ma, T.F.: Remarks on an elliptic equation of Kirchhoff type. Nonlinear Anal. 63, 1967–1977 (2005)
Gudi, T.: Finite element method for a nonlocal problem of Kirchhoff type. SIAM J. Numer. Anal. 50(2), 657–668 (2012)
Gronwall, T.H.: Note on the derivatives with respect to a parameter of the solutions of a system of differential equations. Ann. Math. 20(2), 293–296 (1919)
Dragomir, S.S.: Some Gronwall Type Inequalities and Applications. RGMIA Monographs, Victoria University, Melbourne (2002)
Pani, A.K., Thomée, V., Wahlbin, L.B.: Numerical methods for hyperbolic and parabolic integro-differential equations. J. Integr. Equ. Appl. 4(4), 533–584 (2002)
Lions, J.L.: Quelques methodes de résolution des problémes aux limites non linéaires, Dunod (1969)
Coddington, E.A., Levinson, N.: Theory of Ordinary Differential Equations. McGraw Hill, New York (1965)
Sharma, N., Sharma, K.K.: Unconditionally stable numerical method for a nonlinear partial integro-differential equation. Comput. Math. Appl. 67(1), 62–76 (2014)
Gudi, T., Gupta, H.S.: A fully discrete \(C^0\) interior penalty Galerkin approximation of the extended Fisher–Kolmogorov equation. J. Comput. Appl. Math. 247, 1–16 (2013)
Chipot, M.: Elements of Nonlinear Analysis. Birkhauser Advanced Texts, Berlin (2000)
Wheeler, M.F.: A Priori \(l_2\) error estimates for Galerkin approximations to parabolic partial differential equations. SIAM J. Numer. Anal. 10(4), 723–759 (1973)
Rannacher, R., Scott, R.: Some optimal error estimates for piecewise linear finite element approximation. Math. Compet. 38, 437–445 (1982)
Govaerts, W., Pryce, J.D.: Block elimination with one refinement solves bordered linear system accurately. BIT 30, 490–507 (1990)
Pani, A.K., Fairweather, G.: \(H^1\)-Galerkin mixed finite element methods for parabolic partial integro-differential equations. IMA J. Numer. Anal. 22, 231–252 (2002)
Pani, A.K.: An \(H^1\)-Galerkin mixed finite element methods for parabolic partial differential equations. SIAM J. Numer. Anal. 35(2), 712–727 (1998)
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Srivastava, V., Chaudhary, S., Kumar, V.V.K.S. et al. Fully discrete finite element scheme for nonlocal parabolic problem involving the Dirichlet energy. J. Appl. Math. Comput. 53, 413–443 (2017). https://doi.org/10.1007/s12190-015-0975-6
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12190-015-0975-6