Document Type : Research Paper

Author

School of Mathematics and Statistics, Chongqing Three Gorges University, Chongqing 404000, P.R.China

Abstract

In this paper, a positive definite semi-discrete mixed finite element method was presented for two-dimensional parabolic equations. In the new positive definite systems, the gradient equation and flux equations were separated from their scalar unknown equations.  Also, the existence and uniqueness of the semi-discrete mixed finite element solutions were proven. Error estimates were also obtained for the semi-discrete schemes.Finally, a numerical example was presented to show theoretical results.

Keywords

Main Subjects

[1] Zhang and C. Li, “Superconvergence of finite element approximations to parabolic and hyperbolic integro-differential equations”, Northeastern Math., Vol. 3, pp. 279‒288, (2001).
[2] K. Pani and T. E. Peterson, “Finite Element Methods with numerical quadrature for parabolic integrodifferential equations”, SIAM J. Numer. Anal., Vol. 33, pp. 1084‒1105, (1996).
[3] K. Pani and R. K. Sinha, “Error estimates for semidiscrete Galerkin approximation to a time dependent parabolic integro-differential equation with nonsmooth data”, Calcolo, Vol. 37, pp. 181‒205, (2000).
[4] Zhang and Z. Lu, “A V-cycle multigrid method for a viscoelastic fluid flow satisfying an oldroyd-B-type constitutive equation”, Numer. Anal. Appl., Vol. 12, pp. 69‒78, (2008).
[5] Rajen K. Sinha, Richard E. Ewing and Raytcho D. Lazarov, “Mixed finite element approximations of parabolic integro-differential equations with nonsmooth initial data”, SIAM J. Numer. Anal., 47, pp. 3269‒3292, (2009).
[6] Guo and H. Rui, “Least-squares Galerkin procedures for parabolic integro-differential equations”, Appl. Math. Comp., Vol. 150, pp. 749‒762, (2004).
[7] 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., Vol. 1, pp. 830‒844, (2009).
[8] 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. Engng., Vol. 57, pp. 193‒209, (2003).
[9] Chen, H. Liu and S. Liu, “Analysis of two-grid methods for reaction-diffusion equations by expanded mixed finite element methods”, Int. J. Numer. Meth. Engng., Vol. 69, pp. 408‒422, (2007).
[10] R. Li and Q. Li, “Finite volume element methods for nonlinear parabolic integro-differential problems”, J. KSIAM, Vol. 7, pp. 35‒49, (2003).
[11] X. Chen, “Expanded mixed element methods for linear second-order elliptic problems (I)”, RAIRO Model. Math. Anal. Numer., Vol. 32, pp. 479‒499, (1998).
[12] X. Chen, “Expanded mixed element methods for quasilinear second-order elliptic problems (II)”, RAIRO Model. Math. Anal. Numer., Vol. 32, pp. 501‒420, (1998).
[13] X. Chen, “Analysis of expanded mixed methods for fourth-order elliptic problems”, Numer. Methods Partial Differential Equations, Vol. 13, pp. 483‒503, (1997).
[14] P. Yang, “A splitting positive definite mixed element method for miscible displacement of compressible flow in porous media”, Numer. Methods Partial Differential Equations, Vol. 17, pp. 229‒249, (2001).
[15] Liu, H. Li, J. F. Wang and S. He, “Splitting positive definite mixed element methods for pseudo-hyperbolic equations”, Numer. Methods Partial Differential Equations, Vol. 28, pp. 670‒688, (2012).
[16] S. Zhang and D. P. Yang, “A splitting positive definite mixed element method for second-order hyperbolic equations”, Numer. Methods Partial Differential Equations, Vol. 25, pp. 622‒636, (2009).
CAPTCHA Image