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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.

Abstract

The optimal rate of convergence of the wave equation in both the energy and the -norms using continuous Galerkin method is well known. We exploit this technique and design a fully discrete scheme consisting of coupling the nonstandard finite difference method in the time and the continuous Galerkin method in the space variables. We show that, for sufficiently smooth solution, the maximal error in the -norm possesses the optimal rate of convergence where is the mesh size and is the time step size. Furthermore, we show that this scheme replicates the properties of the exact solution of the wave equation. Some numerical experiments should be performed to support our theoretical analysis.