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

الملخص EN

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.