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; ？) 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. 1. Introduction In recent years, many researchers have studied some numerical methods for fourth-order elliptic equations [1–6], fourth-order parabolic equations [5–9], fourth-order wave equations [10, 11], and so on. Chen [1] proposed and analyzed an expanded mixed finite element method for fourth-order elliptic problems. In [2], Chen et al. studied an anisotropic nonconforming element for fourth-order elliptic singular perturbation problem. In [3–6], some mixed finite element (MFE) methods were studied for fourth-order linear/nonlinear elliptic equations. In [7], the FE method was studied for nonlinear Cahn-Hilliard equation. The optimal-order error estimates were obtained in -norm by means of an FE biharmonic projection approximation. In [12], the MFE methods were studied for solving a fourth-order nonlinear reaction diffusion equation. In [8], an -Galerkin MFE method was studied for solving the fourth-order parabolic partial differential equations. Optimal error estimates were derived for both semidiscrete and fully discrete schemes for problems in one space dimension, and error estimates were derived for semidiscrete scheme for several space dimensions, and the stability for fully discrete scheme was proved by the iteration method. In [9], the FE method was studied for fourth-order nonlinear parabolic equation. He et al. [10] studied and analyzed the explicit/implicit MFE methods for a class of fourth-order wave equations. Shi and Peng [13] studied the finite element
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