References
- N. Bressan and A. Quarteroni. An implicit/explicit spectral method for Burgers’ equation. Calcolo, 23(3):265–284, 1987. http://dx.doi.org/10.1007/BF02576532.
- F. Brezzi and M. Fortin. Mixed and Hybrid Finite Element Methods. Springer, Berlin, 1991.
- J. Cadwell, P. Wanless and A.E. Cook. A finite element approach to Burgers’ equation. Appl. Math. Model., 5(3):189–193, 1981. http://dx.doi.org/10.1016/0307-904X(81)90043-3.
- Y. Chen, Y. Huang and D. Yu. A two-grid method for expanded mixed finite-element solution of semilinear reaction–diffusion equations. Int. J. Numer. Meth Eng., 57(1):139–209, 2003.
- Y. Chen, P. Luan and Z. Lu. Analysis of two-grid methods for nonlinear parabolic equations by expanded mixed finite element methods. Adv. Appl. Math. Mech., 1(1):1–15, 2009.
- P.G. Ciarlet. The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam, 1978.
- C.N. Dawson and M.F. Wheeler. Two-grid method for mixed finite element approximations of non-linear parabolic equations. Contemp. Math., 180(2):191–203, 1994. http://dx.doi.org/10.1090/conm/180/01971.
- J.Jr. Douglas, R.E. Ewing and M.F. Wheeler. The approximation of the pressure by a mixed method in the simulation of miscible displacement. RAIRO Analyse Num′erique, 17(1):17–33, 1983.
- J.Jr. Douglas and J.E. Roberts. Global estimates for mixed finite element methods for second order elliptic equations. Math. Comput., 44(1):39–52, 1985.
- Y. Duan and R. Liu. Global estimates for mixed finite element methods for second order elliptic equations. J. Comput. Appl. Math., 206(1):432–439, 2007. http://dx.doi.org/10.1016/j.cam.2006.08.002.
- C.A.J. Fletcher. A comparison of finite element and difference solutions of the one and two dimensional Burgers’ equations. J. Comp. Phys., 51(1):159–188, 1983. http://dx.doi.org/10.1016/0021-9991(83)90085-2.
- F. Hecht, O. Pironneau, A.L. Hyaric and K. Ohtsuka. FREEFEM + +, version 2.3-3, Laboratoire Jacques-Louis Lions, Paris, 2008.
- X. Hu, X. Feng and P. Huang. A new mixed finite element method based on the Crank–Nicolson scheme for Burgers’ equation, Preprint, 2013.
- P. Huang and A. Abduwali. The modified local Crank–Nicolson method for one-and two-dimensional Burgers’ equations. Comput. Math. Appl., 59(8):2452–2463, 2010. http://dx.doi.org/10.1016/j.camwa.2009.08.069.
- Z. Luo and R. Liu. Mixed finite element analysis and numerical simulation for Burgers’ equation. Math. Numer. Sin., 21(3):257–268, 1999. (in Chinese)
- Z. Qiao, Z. Zhang and T. Tang. An adaptive time-stepping strategy for the molecular beam epitaxy models. SIAM J. Sci. Comput., 33(3):1395–1414, 2011. http://dx.doi.org/10.1137/100812781.
- P.A. Raviart and J.M. Thomas. A mixed finite element method for 2nd order elliptic problems. In P.A. Raviart and J.M. Thomas (Eds.), Mathematics Aspects of the Finite Element Method, volume 66 of Lecture Notes in Math., pp. 292–315, Berlin, 1977. Springer.
- L. Shao, X. Feng and Y. He. The local discontinuous Galerkin finite element method for Burgers’ equation. Math. Comput. Model., 54(11):2943–2954, 2011. http://dx.doi.org/10.1016/j.mcm.2011.07.016.
- F. Shi, J. Yu and K. Li. 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(11):2293–2305, 2011. http://dx.doi.org/10.1080/00207160.2010.534466.
- Z. Weng, X. Feng and P. Huang. A new mixed finite element method based on the Crank–Nicolson scheme for the parabolic problems. Appl. Math. Model., 36(12):5068–5079, 2012. http://dx.doi.org/10.1016/j.apm.2011.12.044.
- Z. Weng, X. Feng and D. Liu. A two-grid stabilized mixed finite element method for semilinear elliptic equations. Appl. Math. Model., 37(10-11):7037–7046, 2013. http://dx.doi.org/10.1016/j.apm.2013.02.016.
- Z. Weng, X. Feng and S. Zhai. Investigations on two kinds of two-grid mixed finite element methods for the elliptic eigenvalue problem. Comput. Math. Appl., 64(8):2635–2646, 2012. http://dx.doi.org/10.1016/j.camwa.2012.07.009.
- Z. Weng, X. Feng and S. Zhai. Analysis of two-grid method for semi-linear elliptic equations by new mixed finite element scheme. Appl. Math. Comput., 219(9):4826–4835, 2013. http://dx.doi.org/10.1016/j.amc.2012.10.108.
- L. Wu and M.B. Allen. A two-grid method for mixed finite-element solutions of reaction–diffusion equations. Numer. Meth. PDEs, 15(3):317–332, 1999.
- W. Wu, X. Feng and D. Liu. The local discontinuous Galerkin finite element method for a class of convection-diffusion equations. Nonlinear Anal. Real World Appl., 14(1):734–752, 2013. http://dx.doi.org/10.1016/j.nonrwa.2012.07.030.
- J. Xu. A novel two-grid method for semilinear equations. SIAM J. Sci. Comp., 15(1):231–237, 1994. http://dx.doi.org/10.1137/0915016.
- J. Xu. Two-grid discretization techniques for linear and non-linear PDEs. SIAM J. Numer. Anal., 33(5):1759–1777, 1996. http://dx.doi.org/10.1137/S0036142992232949.
- Z. Zhang and Z. Qiao. An adaptive time-stepping strategy for the cahn–hilliard equation. Commun. Comput. Phys., 11(1):1261–1278, 2012.