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))2 space, which avoids the use of the classical H(div; Ω) space and reduces the regularity requirement on the gradient solution λ = ∇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 in L2 and H1‐norm for both the scalar unknown u and the diffusion term γ and a priori error estimates in (L2) 2‐norm for its gradient λ and its flux σ (the coefficients times the negative gradient). Finally, we provide some numerical results to illustrate the efficiency of our method.