Estimating the error of a \(H^{1}\)-mixed finite element solution for the Burgers equation

Shafie, Sabarina Binti ; Tran, Thanh

ANZIAM Journal, Tome 56 (2016), / Harvested from Australian Mathematical Society

Résumé

We compute error estimations for a \(H^{1}\)-mixed finite element method for Burgers equation. By using a \(H^{1}\)-mixed finite element method, the problem is reformulated as a system of first order partial differential equations, which allows an approximation of the unknown function and its derivative. Local parabolic and elliptic methods approximate the true errors from the computed solutions; the so-called a posteriori error estimates. Numerical experiments show that the error estimations converge to the true errors. References S. Adjerid, J. E. Flaherty, and Y. J. Wang. A posteriori error estimation with finite element methods of lines for one-dimensional parabolic systems. Numer. Math., 65(1):1–21, 1993. doi:10.1007/BF01385737 J. M. Burgers. The Nonlinear Diffusion Equation: Asymptotic Solutions and Statistical Problems. Springer, 1974. doi:10.1007/978-94-010-1745-9 A. K. Pani. An \(H^1\)-Galerkin mixed finite element method for parabolic partial differential equations. SIAM J. Numer. Anal., 35(2):712–727, 1998. doi:10.1137/S0036142995280808 A. K. Pany, N. Nataraj and S. Singh. A new mixed finite element method for Burgers' equation. J. Appl. Math. Comput., 23(1):43–55, 2007. doi:10.1007/BF02831957 T. Tran and T.-B. Duong. A complete analysis for some a posteriori error estimates with the finite element method of lines for a nonlinear parabolic equation. Numer. Funct. Anal. Optim., 23(7–8):891–909, 2002. doi:10.1081/NFA-120016275 T. Tran and T.-B. Duong. A posteriori error estimates with the finite element method of lines for a Sobolev equation. Numer. Methods P D. E., 21(3):521–535, 2005. doi:10.1002/num.20045 Z. Cai, R. Lazarov, T. A. Manteuffel and S. F. McCormick. First-order system least squares for second-order partial differential equations: Part I. SIAM J. Numer. Anal., 31(6):1785–1799, 1994. doi:10.1137/0731091 Z. Cai, T. A. Manteuffel and S. F. McCormick. First-order system least squares for second-order partial differential equations: Part II. SIAM J. Numer. Anal., 34(2):425–454, 1997. doi:10.1137/S0036142994266066 A. I. Pehlivanov, G. F. Carey and R. D. Lazarov. Least-squares mixed finite elements for second-order elliptic problems. SIAM J. Numer. Anal., 31(5):1368–1377, 1994. doi:10.1137/0731071 A. I. Pehlivanov, G. F. Carey, R. D. Lazarov and Y. Shen. Convergence analysis of least-squares mixed finite elements. Computing, 51(2):111–123, 1993. doi:10.1007/BF02243846 Tomasz Dlotko. The one-dimensional Burgers' equation; existence, uniqueness and stability. Zeszyty Nauk. Uniw. Jagiellon. Prace Mat., (23):157–172, 1982. J.-L. Lions. Quelques methodes de resolution des problemes aux limites non lineaires. Dunod; Gauthier-Villars, Paris, 1969.

Détail

Publié le : 2016-01-01
DOI : https://doi.org/10.21914/anziamj.v56i0.9356