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

We propose and analyze a new numericalmethod, called a couplingmethod based on a new expandedmixed 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 FEmethod and approximate the other one by a new EMFEmethod. We find that the new EMFEmethod’s gradient belongs to the simple square integrable (L(Ω))2 space, which avoids the use of the classicalH(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 andH-norm for both the scalar unknown u and the diffusion term γ and a priori error estimates in (L)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.