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.


Remark 1.
From the new coupling weak formulation (5a)-(5d), we can find that the gradient belongs to the weaker square integrable ( 2 (Ω)) 2 space taking the place of the classical H(div; Ω) space.It is easy to see that our method reduces the regularity requirement on the gradient solution  = ∇.Remark 2. In the semidiscrete coupling scheme (6a)-(6d), the MFE space ( ℎ , W ℎ ) based on the FE pair  1 −  2 0 is chosen as follows: From [30,31], we know that MFE space ( ℎ , W ℎ ) satisfies the discrete LBB condition.

Semidiscrete Error Estimates
For deriving some a priori error estimates of our method, we first introduce the projection operator P ℎ and the new expanded mixed elliptic projection operator R ℎ associated with the coupled equations.

Lemma 5. One now defines a linear projection operator P
and then      − P ℎ Let (R ℎ , R ℎ , R ℎ ) : [0, ] →  ℎ × W ℎ × W ℎ be given by the following new expanded mixed relations [29] Then the following two important lemmas based on the new expanded mixed projection (10a)-(10c) hold.Lemma 6.There is a constant  > 0 independent of the spatial mesh parameter ℎ such that In [29], we can obtain the detailed proof for Lemmas 6-7.

Remark 9.
From Theorem 8, we can see that that is to say, a priori error estimates in  2 and  1 -norms for the scalar unknown  are optimal.
Remark 11.From Theorem 10, we can see that that is to say, a priori error estimates in  2 and  1 -norm for the diffusion term  are optimal.

Fully Discrete Error Estimates
In the following analysis, we will derive a priori error estimates based on fully discrete backward Euler scheme.Let 0 =  0 <  1 <  2 < ⋅ ⋅ ⋅ <   =  be a given partition of the time interval [0, ] with step length Δ = / and nodes   = Δ, for some positive integer .For a smooth function  on [0, ], define   = (  ).

Numerical Results
In this section, we would like to give some numerical results for the coupling method of EMFE method and FE method proposed and analyzed in this paper.We consider the following initial-boundary value problem of fourth-order parabolic system: where Ω = [0, 1] × [0, 1],  = (0, 1], () = 1 +  2 ,  = 1, and (x, ) is chosen so that the exact solution for the scalar unknown function is  (x, ) =  −2 sin ( 1 ) sin ( 2 ) , Dividing the domain Ω into the triangulations of mesh size ℎ uniformly and choosing the piecewise linear space  ℎ with index  = 1 and ℎ = √ 2Δ = √ 2/8, √ 2/16, √ 2/32, √ 2/64, we obtain the optimal a priori error estimates in  2 and  1 -norm for the scalar unknown  in Table 1.Similarly, from Table 2, we can find that both ‖ −  ℎ ‖  ∞ ( 1 (Ω)) and Advances in Mathematical Physics    From the convergence results in Tables 1-3 and Figures 1-8, we can find that the numerical results verify our theoretical analysis.

Concluding Remarks
In this article, we study a new coupling method of new EMFE scheme [29] and FE scheme for fourth-order partial differential equation of parabolic type.The new EMFE method's gradient belongs to the simple square integrable ( 2 (Ω)) 2 space; therefore, the regularity requirement on the gradient solution  = ∇ is reduced.We obtain the optimal priori error estimates in  2 and  1 -norm for both the scalar unknown  and the diffusion term  and the priori error estimates in ( 2 ) 2 -norm for its gradient  and its flux .Finally, a numerical example is provided to verify the efficiency of our methods.At the same time, we apply the new coupling method based on new EMFE method and FE method to solving the fourth-order eigenvalue problems, which will be shown in another article.
In the near future, we will study the new expanded mixed finite element method for other partial differential equations, such as fourth-order wave equations.Moreover, we will apply some new techniques based on the combination of two-grid methods [36,37] and new (expanded) mixed methods [29][30][31][32]38] for solving fourth-order nonlinear elliptic equation and fourth-order nonlinear parabolic equations.

Table 2 :
2 and  1 -norms and convergence order for .