A Coupling Method of New EMFE and FE for Fourth-Order Partial Differential Equation of Parabolic Type. (18th April 2013) - Liu, Yang; Li, Hong; Fang, Zhichao; He, Siriguleng; Wang, Jinfeng
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( L 2 ( Ω ) ) 2 space, 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 inL 2 andH 1 -norm for both the scalar unknownu and the diffusion term γ and a priori error estimates in( L 2 ) 2 -norm for its gradientλ and its fluxσ (the coefficients times the negative gradient). Finally, we provide some numerical results to illustrate the efficiency of our method.