Article
Full entry |
PDF
(1.9 MB)
Feedback
PDF
(1.9 MB)
Feedback
Keywords:
Burgers' equation; mixed finite element method; stable conforming finite element; Crank-Nicolson scheme; inf-sup condition
Burgers' equation; mixed finite element method; stable conforming finite element; Crank-Nicolson scheme; inf-sup condition
Summary:
In this paper, a new mixed finite element method is used to approximate the solution as well as the flux of the 2D Burgers' equation. Based on this new formulation, we give the corresponding stable conforming finite element approximation for the $P_0^2-P_1$ pair by using the Crank-Nicolson time-discretization scheme. Optimal error estimates are obtained. Finally, numerical experiments show the efficiency of the new mixed method and justify the theoretical results.
References:
[1] Adams, R. A.: Sobolev Spaces. Pure and Applied Mathematics 65 Academic Press, New York (1975). MR 0450957 | Zbl 0314.46030
[2] Bateman, H.: Some recent researches on motion of fluids. Mon. Weather Rev. 43 (1915), 163-170. DOI 10.1175/1520-0493(1915)43<163:SRROTM>2.0.CO;2
[3] Bressan, N., Quarteroni, A.: An implicit/explicit spectral method for Burgers' equation. Calcolo 23 (1986), 265-284. DOI 10.1007/BF02576532 | MR 0897632 | Zbl 0691.65081
[4] Cadwell, J., Wanless, P., Cook, A. E.: A finite element approach to Burgers' equation. Appl. Math. Modelling 5 (1981), 189-193. DOI 10.1016/0307-904X(81)90043-3 | MR 0626869
[5] Chen, H., Jiang, Z.: A characteristics-mixed finite element method for Burgers' equation. J. Appl. Math. Comput. 15 (2004), 29-51. DOI 10.1007/BF02935745 | MR 2043967 | Zbl 1053.65083
[6] Ciarlet, P. G.: The Finite Element Method for Elliptic Problems. Studies in Mathematics and Its Applications. Vol. 4 North-Holland Publishing Company, Amsterdam (1978). MR 0520174 | Zbl 0383.65058
[7] Crank, J., Nicolson, P.: A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type. Proc. Camb. Philos. Soc. 43 (1947), 50-67 Reprint in Adv. Comput. Math. 6 (1996), 207-226. DOI 10.1017/S0305004100023197 | MR 0019410 | Zbl 0866.65054
[8] Duan, Y., Liu, R.: Lattice Boltzmann model for two-dimensional unsteady Burgers' equation. J. Comput. Appl. Math. 206 (2007), 432-439. DOI 10.1016/j.cam.2006.08.002 | MR 2337455 | Zbl 1115.76064
[9] Fletcher, C. A. J.: A comparison of finite element and finite difference solutions of the one- and two-dimensional Burgers' equations. J. Comput. Phys. 51 (1983), 159-188. DOI 10.1016/0021-9991(83)90085-2 | MR 0713944 | Zbl 0525.65077
[10] He, Y.: Two-level method based on finite element and Crank-Nicolson extrapolation for the time-dependent Navier-Stokes equations. SIAM J. Numer. Anal. 41 (2003), 1263-1285. DOI 10.1137/S0036142901385659 | MR 2034880 | Zbl 1130.76365
[11] He, Y., Li, J.: Convergence of three iterative methods based on the finite element discretization for the stationary Navier-Stokes equations. Comput. Methods Appl. Mech. Eng. 198 (2009), 1351-1359. DOI 10.1016/j.cma.2008.12.001 | MR 2497612 | Zbl 1227.76031
[12] He, Y., Sun, W.: Stability and convergence of the Crank-Nicolson/Adams-Bashforth scheme for the time-dependent Navier-Stokes equations. SIAM J. Numer. Anal. 45 (2007), 837-869. DOI 10.1137/050639910 | MR 2300299 | Zbl 1145.35318
[13] He, Y., Sun, W.: Stabilized finite element method based on the Crank-Nicolson extrapolation scheme for the time-dependent Navier-Stokes equations. Math. Comput. 76 (2007), 115-136. DOI 10.1090/S0025-5718-06-01886-2 | MR 2261014 | Zbl 1129.35004
[14] Hecht, F., Pironneau, O., Hyaric, A. Le, Ohtsuka, K.: FREEFEM++, version 2.3-3, 2008. Software available at http://www.freefem.org</b>
[15] Heywood, J. G., Rannacher, R.: Finite-element approximation of the nonstationary Navier-Stokes problem. IV. Error analysis for second-order time discretization. SIAM J. Numer. Anal. 27 (1990), 353-384. DOI 10.1137/0727022 | MR 1043610 | Zbl 0694.76014
[16] Huang, P., Abduwali, A.: The modified local Crank-Nicolson method for one- and two-dimensional Burgers' equations. Comput. Math. Appl. 59 (2010), 2452-2463. DOI 10.1016/j.camwa.2009.08.069 | MR 2607949 | Zbl 1193.65157
[17] Johnston, H., Liu, J. G.: Accurate, stable and efficient Navier-Stokes solvers based on explicit treatment of the pressure term. J. Comput. Phys. 199 (2004), 221-259. DOI 10.1016/j.jcp.2004.02.009 | MR 2081004 | Zbl 1127.76343
[18] Luo, Z., Liu, R.: Mixed finite element analysis and numerical simulation for Burgers equation. Math. Numer. Sin. 21 (1999), 257-268 Chinese. MR 1762984 | Zbl 0933.65117
[19] Pany, A. K., Nataraj, N., Singh, S.: A new mixed finite element method for Burgers' equation. J. Appl. Math. Comput. 23 (2007), 43-55. DOI 10.1007/BF02831957 | MR 2282449 | Zbl 1124.65095
[20] Shang, Y.: Initial-boundary value problems for a class of generalized KdV-Burgers equations. Math. Appl. 9 (1996), 166-171 Chinese. MR 1405073 | Zbl 0937.35164
[21] Shao, L., Feng, X., He, Y.: The local discontinuous Galerkin finite element method for Burger's equation. Math. Comput. Modelling 54 (2011), 2943-2954. DOI 10.1016/j.mcm.2011.07.016 | MR 2841837 | Zbl 1235.65115
[22] Shi, F., Yu, J., Li, K.: A new stabilized mixed finite-element method for Poisson equation based on two local Gauss integrations for linear element pair. Int. J. Comput. Math. 88 (2011), 2293-2305. DOI 10.1080/00207160.2010.534466 | MR 2818083 | Zbl 1241.65091
[23] Weng, Z., Feng, X., Huang, P.: A new mixed finite element method based on the Crank-Nicolson scheme for the parabolic problems. Appl. Math. Modelling 36 (2012), 5068-5079. DOI 10.1016/j.apm.2011.12.044 | MR 2930402 | Zbl 1252.65170
Search
Partner of
