AM  Vol.2 No.8 , August 2011
A Posteriori Error Estimate for Streamline Diffusion Method in Soving a Hyperbolic Equation
Abstract: In this article, we use streamline diffusion method for the linear second order hyperbolic initial-boundary value problem. More specifically, we prove a posteriori error estimates for this method for the linear wave equation. We observe that this error estimates make finite element method increasingly powerful rather than other methods.
Cite this paper: nullD. Rostamy and F. Zabihi, "A Posteriori Error Estimate for Streamline Diffusion Method in Soving a Hyperbolic Equation," Applied Mathematics, Vol. 2 No. 8, 2011, pp. 981-986. doi: 10.4236/am.2011.28135.

[1]   M. Ainsworth and J. Tinsley Oden, “A Posteriori Error Estimation in Finite Element Analysis,” Wiley-Interscience, New York, 2000.

[2]   M. Asadzadeh, “A Posteriori Error Estimates for the Fokker-Planck and Fermi Pencil Beam Equations,” Mathematical Methods in the Applied Sciences, Vol. 10, No. 3, 2000, pp. 737-769. doi:10.1142/S0218202500000380

[3]   E. H. Gergouli, O. Lakkis and C. Makridakis, “A Posteriori L1(L2)-Error Bounds in Finite Element Approximation of the Wave Equation,” arXiv:1003.3641v1[math. NA], 2010, pp. 1-17.

[4]   C. Johnson, “Numerical Solutions of Partial Differential Equations by the Finite Element Method,” Cambridge University, Cambridge, 1987.

[5]   C. Johnson, “Discontinous Galerkin Finite Element Methods for Second Order Hyperbolic Problems,” Computer Methods in Applied Mechanics and Engineering, Vol. 107, No. 5, 1993, pp. 117-129. doi:10.1016/0045-7825(93)90170-3

[6]   M. Asadzadeh, “Streamline Diffusion Methods for the Vlasov-Poisson Equations. RAIRO Math,” Modelling and Numerical Analysis, Vol. 24, No. 3, 1990, pp. 177-196.

[7]   M. Asadzadeh and P. Kowalczyk, “Convergence of Streamline Diffusion Methods for the Vlasov-Poisson-Fokker-Planck System,” Numerical Methods for Partial Differential Equations, Vol. 21, No. 2, 2005, pp. 472-495. doi:10.1002/num.20044

[8]   K. Eriksson and C. Johnson, “Adaptive Streamline Diffusion Finite Element Methods for Stationary Convection-Diffusion Problems,” Mathematics of Computation, Vol. 60, No. 4, 1993, pp. 167-188. doi:10.1090/S0025-5718-1993-1149289-9

[9]   S. C. Brenner and L. R. Scott, “The Mathematical Theory of Finite Element Method,” Springer-Verlag, New York, 1994.

[10]   F. Dubois and P. G. Le Floch, “Boundary Conditions for Nonlinear Hyperbolic Systems of Conservation Laws,” Journal of Differential Equations, Vol. 71, No. 3, 2001, pp. 93-122.

[11]   C. Fuhrer and R. Rannacher, “An Adaptive Streamline Diffusion Finite Element Method for Hyperbolic Conservation Laws,” East-West Journal of Numerical Mathematics, Vol. 5, No. 2, 1997, pp. 145-162.

[12]   R. Codina,” Finite Element Approximation of the Hyperbolic Wave Equation in Mixed Form,” Computer Methods in Applied Mechanics and Engineering, Vol. 197, No. 6, 2008, pp. 1305-1322. doi:10.1016/j.cma.2007.11.006

[13]   L. Haws, “Symmetric Greens Functions for Certain Hyperbolic Problems,” Computers & Mathematics with Applications, 21, 1991, pp. 65-78. doi:10.1016/0898-1221(91)90216-Q

[14]   N, Iraniparast, “A Boundary Value Problem for the Wave Equation,” International Journal of Mathematics and Mathematical Sciences, Vol. 22, No. 4, 1999, pp. 835-845. doi:10.1155/S0161171299228359

[15]   T. Kalmenov, “On the Spectrum of a Selfadjoint Problem for the Wave Equation,” Akad. Nauk. Kazakh. SSR, Vestnik, Vol. 1, No. 3, 1983, pp. 63-66.

[16]   R. A. Adams, “Sobolev Spaces,” Academic Press, New York, 1975.

[17]   A. Shermenew, “Nonlinear Wave Equation in Special Coordinates,” Journal of Nonlinear Mathematical Physics, Vol. 11, No. 2, 2004, pp. 110-115. doi:10.2991/jnmp.2004.11.s1.14

[18]   E. Burman, “Adaptive Finite Element Methods for Compressible Two-Phase Flow,” Mathematical Methods in the Applied Sciences, Vol. 10, No. 2, 2000, pp. 963-989. doi:10.1016/S0218-2025(00)00049-5

[19]   C. Johnson and A. Szepessy, “Adaptive Finite Element Methods for Conservation Laws Based on a Posteriori Error Estimates,” Communications on Pure and Applied Mathematics, Vol. 48, No. 3, 1995, pp. 199-234. doi:10.1002/cpa.3160480302

[20]   R. Sandboge, “Adaptive Finite Element Methods for Systems of Reaction-Diffusion Equations,” Computer Methods in Applied Mechanics and Engineering, Vol. 166, No. 3, 1998, pp. 309-328. doi:10.1016/S0045-7825(98)00093-0

[21]   P. G. Ciarlet, “The Finite Element Method for Elliptic Problems,” Amesterdam, North Holland, 1987.