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Advances in Mathematical Physics
Volume 2013 (2013), Article ID 787891, 14 pages
http://dx.doi.org/10.1155/2013/787891
Research Article

A Coupling Method of New EMFE and FE for Fourth-Order Partial Differential Equation of Parabolic Type

1School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China
2School of Statistics and Mathematics, Inner Mongolia University of Finance and Economics, Hohhot 010070, China

Received 29 November 2012; Accepted 11 March 2013

Academic Editor: B. G. Konopelchenko

Copyright © 2013 Yang Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We propose and analyze a new numerical method, called a coupling method based on a new expanded mixed finite element (EMFE) and finite element (FE), for fourth-order partial differential equation of parabolic type. We first reduce the fourth-order parabolic equation to a coupled system of second-order equations and then solve a second-order equation by FE method and approximate the other one by a new EMFE method. We find that the new EMFE method’s gradient belongs to the simple square integrable space, which avoids the use of the classical H(div; Ω) space and reduces the regularity requirement on the gradient solution . For a priori error estimates based on both semidiscrete and fully discrete schemes, we introduce a new expanded mixed projection and some important lemmas. We derive the optimal a priori error estimates in and -norm for both the scalar unknown and the diffusion term γ and a priori error estimates in -norm for its gradient and its flux (the coefficients times the negative gradient). Finally, we provide some numerical results to illustrate the efficiency of our method.