Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

A Superconvergent discontinuous Galerkin method for Volterra integro-differential equations, smooth and non-smooth kernels


Author: Kassem Mustapha
Journal: Math. Comp. 82 (2013), 1987-2005
MSC (2010): Primary 65-XX
DOI: https://doi.org/10.1090/S0025-5718-2013-02689-0
Published electronically: April 18, 2013
MathSciNet review: 3073189
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study the numerical solution for Volerra integro-differential equations with smooth and non-smooth kernels. We use an $ h$-version discontinuous Galerkin (DG) method and derive nodal error bounds that are explicit in the parameters of interest. In the case of non-smooth kernel, it is justified that the start-up singularities can be resolved at superconvergence rates by using non-uniformly graded meshes. Our theoretical results are numerically validated in a sample of test problems.


References [Enhancements On Off] (What's this?)

  • 1. H. Brunner, Collocation Method for Volterra Integral and Related Functional Differential Equations, Cambridge University Press, Cambridge, 2004. MR 2128285 (2005k:65002)
  • 2. H. Brunner, A. Pedas and G. Vainikko, The piecewise polynomial collocation methods for linear Volterra integrodifferential equations with weakly singular kernels, SIAM J. Numer. Anal., 39 (2001), 957-982. MR 1860452 (2002f:65190)
  • 3. H. Brunner and D. Schötzau, $ hp$-Discontinuous Galerkin time stepping for Volterra integrodifferential equations, SIAM J. Numer. Anal., 44 (2006), 224-245. MR 2217380 (2007h:65150)
  • 4. M. Delfour and W. Hager and F. Trochu, Discontinuous Galerkin methods for ordinary differential equations, Math. Comp., 36 (1981), 455-473. MR 606506 (82b:65066)
  • 5. K. Eriksson and C. Johnson and Thomée, Time discretization of parabolic problems by the discontinuous Galerkin method, RAIRO Modél. Math. Anal. Numér., 19 (1985), 611-643. MR 826227 (87e:65073)
  • 6. D. Estep, A posteriori error bounds and global error control for approximation of ordinary differential equations, SIAM J. Numer. Anal., 32 (1995), 1-48. MR 1313704 (96i:65049)
  • 7. Y. J. Jiang, On spectral methods for Volterra-type Integro-differential equations, J. Comput. Appl. Math., 230 (2009), 333-340. MR 2532327 (2010d:45011)
  • 8. C. Johnson, Error estimates and adaptive time-step control for a class of one-step methods for stiff ordinary differential equations, SIAM J. Numer. Anal., 25 (1988), 908-926. MR 954791 (90a:65160)
  • 9. S. Larsson, V. Thomée and L. Wahlbin, Numerical solution of parabolic integro-differential equations by the discontinuous Galerkin method, Math. Comp., 67 (1998), 45-71. MR 1432129 (98d:65168)
  • 10. P. Lesaint and P.A. Raviart, On a finite element method for solving the neutron transport equation in Mathematical Aspects of Finite Elements in Partial Differential Equations (Madison, 1974), editor: C. de Boor, Academic Press, New York, (1974), 89-145. MR 0658142 (58:31918)
  • 11. W. Mclean and K. Mustapha, A second-order accurate numerical method for a fractional wave equation, Numer. Math., 105 (2007), 481-510. MR 2266834 (2008d:65097)
  • 12. K. Mustapha, A Petrov-Galerkin method for integro-differential equations with a memory term, ANZIAM J., 50 (2008), 610-624. MR 2470608 (2009j:65373)
  • 13. K. Mustapha and W. McLean, Discontinuous Galerkin method for an evolution equation with a memory term of positive type, Math. Comp., 78 (2009), 1975-1995. MR 2521275 (2010f:65190)
  • 14. K. Mustapha, H. Brunner, H. Mustapha and D. Schötzau, An hp-version discontinuous Galerkin method for integro-differential equations of parabolic type, SIAM J. Numer. Anal., 49 (2011), 1369-1396. MR 2817543
  • 15. K. Mustapha and H. Mustapha, A second-order accurate numerical method for a semilinear integro-differential equation with a weakly singular kernel, IMA J. Numer. Anal., 30 (2010), 555-578. MR 2608473 (2011d:65223)
  • 16. A. Pani, G. Fairweather and R. Fernandes, ADI orthogonal spline collocation methods for parabolic partial integro-differential equations, IMA J. Numer. Anal., 30 (2010), 248-276. MR 2580558 (2011e:65211)
  • 17. A. Pani and S. Yadav, An $ hp$-local discontinuous Galerkin method for parabolic integro-differential equations, J. Sci. Comput., 46 (2011), 71-99. MR 2753252 (2012b:65143)
  • 18. W.H. Reed and T.R. Hill, Triangular mesh methods for the neutron transport equation. Los Alamos Scientific Laboratory, LA-UR-73-479, 1973.
  • 19. D. Schötzau and C. Schwab, Time discretization of parabolic problems by the $ hp$-version of the discontinuous Galerkin finite element method, SIAM J. Numer. Anal., 38 (2000), 837-875. MR 1781206 (2001i:65107)
  • 20. D. Schötzau and C. Schwab, An hp a-priori error analysis of the DG time-stepping method for initial value problems, Calcolo, 37 (2000), 207-232. MR 1812787 (2001k:65126)
  • 21. T. Tang, A note on collocation methods for Volterra integro-differential equations with weakly singular kernels, IMA J. Numer. Anal., 13 (1993), 93-99. MR 1199031 (93k:65111)
  • 22. T. Tang, X. Xu and J. Chen, On spectral methods for Volterra integral equations and the convergence analysis, J. Comput. Math., 26 (2008), 825-837. MR 2464738 (2010c:65256)
  • 23. Y. Wei and Y. Chen, Convergence analysis of the spectral methods for weakly singular Volterra integro-differential equations with smooth solutions, Adv. Appl. Math. Mech., 4 (2012), 1-20.
  • 24. V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Springer Ser. Comput. Math. 25, Springer-Verlag, Berlin, 2006. MR 2249024 (2007b:65003)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 65-XX

Retrieve articles in all journals with MSC (2010): 65-XX


Additional Information

Kassem Mustapha
Affiliation: Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, 31261, Saudi Arabia.
Email: kassem@kfupm.edu.sa

DOI: https://doi.org/10.1090/S0025-5718-2013-02689-0
Keywords: Integro-differential equation, weakly singular kernel, smooth kernel, DG time-stepping, error analysis, variable time steps
Received by editor(s): January 22, 2011
Received by editor(s) in revised form: November 18, 2011, and January 31, 2012
Published electronically: April 18, 2013
Additional Notes: Support of the KFUPM through the project SB101020 is gratefully acknowledged.
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society