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

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(L2(Ω))2space, which avoids the use of the classicalH(div; Ω) space and reduces the regularity requirement on the gradient solutionλ=∇u. 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 inL2andH1-norm for both the scalar unknownuand the diffusion termγand a priori error estimates in(L2)2-norm for its gradientλand its fluxσ(the coefficients times the negative gradient). Finally, we provide some numerical results to illustrate the efficiency of our method.