Table of Contents Author Guidelines Submit a Manuscript
Journal of Applied Mathematics
Volume 2013 (2013), Article ID 520219, 9 pages
http://dx.doi.org/10.1155/2013/520219
Research Article

Optimal Rate of Convergence for a Nonstandard Finite Difference Galerkin Method Applied to Wave Equation Problems

Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 0002, South Africa

Received 6 August 2013; Revised 11 November 2013; Accepted 14 November 2013

Academic Editor: Song Cen

Copyright © 2013 Pius W. M. Chin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. D. A. French and J. W. Schaeffer, “Continuous finite element methods which preserve energy properties for nonlinear problems,” Applied Mathematics and Computation, vol. 39, no. 3, pp. 271–295, 1990. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. D. A. French and T. E. Peterson, “A continuous space-time finite element method for the wave equation,” Mathematics of Computation, vol. 65, no. 214, pp. 491–506, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. R. T. Glassey, “Convergence of an energy-preserving scheme for the Zakharov equations in one space dimension,” Mathematics of Computation, vol. 58, no. 197, pp. 83–102, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. R. Glassey and J. Schaeffer, “Convergence of a second-order scheme for semilinear hyperbolic equations in 2 + 1 dimensions,” Mathematics of Computation, vol. 56, no. 193, pp. 87–106, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. W. Strauss and L. Vazquez, “Numerical solution of a nonlinear Klein-Gordon equation,” Journal of Computational Physics, vol. 28, no. 2, pp. 271–278, 1978. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. T. Dupont, “L2-estimates for Galerkin methods for second order hyperbolic equations,” SIAM Journal on Numerical Analysis, vol. 10, pp. 880–889, 1973. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. G. A. Baker, “Error estimates for finite element methods for second order hyperbolic equations,” SIAM Journal on Numerical Analysis, vol. 13, no. 4, pp. 564–576, 1976. View at Publisher · View at Google Scholar · View at MathSciNet
  8. E. Gekeler, “Linear multistep methods and Galerkin procedures for initial boundary value problems,” SIAM Journal on Numerical Analysis, vol. 13, no. 4, pp. 536–548, 1976. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. C. Johnson, “Discontinuous Galerkin finite element methods for second order hyperbolic problems,” Computer Methods in Applied Mechanics and Engineering, vol. 107, no. 1-2, pp. 117–129, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. G. R. Richter, “An explicit finite element method for the wave equation,” Applied Numerical Mathematics, vol. 16, no. 1-2, pp. 65–80, 1994, A Festschrift to honor Professor Robert Vichnevetsky on his 65th birthday. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. P. W. M. Chin, J. K. Djoko, and J. M.-S. Lubuma, “Reliable numerical schemes for a linear diffusion equation on a nonsmooth domain,” Applied Mathematics Letters, vol. 23, no. 5, pp. 544–548, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. R. E. Mickens, Nonstandard Finite Difference Models of Differential Equations, World Scientific Publishing, River Edge, NJ, USA, 1994. View at MathSciNet
  13. R. Anguelov and J. M.-S. Lubuma, “Contributions to the mathematics of the nonstandard finite difference method and applications,” Numerical Methods for Partial Differential Equations, vol. 17, no. 5, pp. 518–543, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. R. Anguelov and J. M.-S. Lubuma, “Nonstandard finite difference method by nonlocal approximation,” Mathematics and Computers in Simulation, vol. 61, no. 3–6, pp. 465–475, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. S. M. Moghadas, M. E. Alexander, B. D. Corbett, and A. B. Gumel, “A positivity-preserving Mickens-type discretization of an epidemic model,” Journal of Difference Equations and Applications, vol. 9, no. 11, pp. 1037–1051, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. K. C. Patidar, “On the use of nonstandard finite difference methods,” Journal of Difference Equations and Applications, vol. 11, no. 8, pp. 735–758, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. J. L. Lions, E. Magenes, and P. Kenneth, Non-Homogeneous Boundary Value Problems and Applications, vol. 1, Springer, Berlin, Germany, 1972.
  18. R. Uddin, “Comparison of the nodal integral method and nonstandard finite-difference schemes for the Fisher equation,” SIAM Journal on Scientific Computing, vol. 22, no. 6, pp. 1926–1942, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  19. J. Oh and D. A. French, “Error analysis of a specialized numerical method for mathematical models from neuroscience,” Applied Mathematics and Computation, vol. 172, no. 1, pp. 491–507, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet