Hindawi Publishing Corporation Advances in Mathematical Physics Volume 2013, Article ID 787891, 14 pages http://dx.doi.org/10.1155/2013/787891 Research Article A Coupling Method of New EMFE and FE for Fourth-Order Partial Differential Equation of Parabolic Type Yang Liu,1 Hong Li,1 Zhichao Fang,1 Siriguleng He,1 and Jinfeng Wang2 1 2 School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China School of Statistics and Mathematics, Inner Mongolia University of Finance and Economics, Hohhot 010070, China Correspondence should be addressed to Yang Liu; mathliuyang@yahoo.cn and Hong Li; smslh@imu.edu.cn Received 29 November 2012; Accepted 11 March 2013 Academic Editor: B. G. Konopelchenko Copyright © 2013 Yang Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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 2 other one by a new EMFE method. We find that the new EMFE methodβs gradient belongs to the simple square integrable (πΏ2 (Ξ©)) space, which avoids the use of the classical H(div; Ξ©) space and reduces the regularity requirement on the gradient solution π = βπ’. 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 πΏ2 and π»1 -norm for both the scalar unknown π’ and 2 the diffusion term πΎ and a priori error estimates in (πΏ2 ) -norm for its gradient π and its flux π (the coefficients times the negative gradient). Finally, we provide some numerical results to illustrate the efficiency of our method. 1. Introduction In recent years, many researchers have studied some numerical methods for fourth-order elliptic equations [1β6], fourthorder parabolic equations [5β9], fourth-order wave equations [10, 11], and so on. Chen [1] proposed and analyzed an expanded mixed finite element method for fourth-order elliptic problems. In [2], Chen et al. studied an anisotropic nonconforming element for fourth-order elliptic singular perturbation problem. In [3β6], some mixed finite element (MFE) methods were studied for fourth-order linear/nonlinear elliptic equations. In [7], the FE method was studied for nonlinear Cahn-Hilliard equation. The optimalorder error estimates were obtained in πΏ2 -norm by means of an FE biharmonic projection approximation. In [12], the MFE methods were studied for solving a fourth-order nonlinear reaction diffusion equation. In [8], an π»1 -Galerkin MFE method was studied for solving the fourth-order parabolic partial differential equations. Optimal error estimates were derived for both semidiscrete and fully discrete schemes for problems in one space dimension, and error estimates were derived for semidiscrete scheme for several space dimensions, and the stability for fully discrete scheme was proved by the iteration method. In [9], the FE method was studied for fourth-order nonlinear parabolic equation. He et al. [10] studied and analyzed the explicit/implicit MFE methods for a class of fourth-order wave equations. Shi and Peng [13] studied the finite element methods for fourth-order eigenvalue problems on anisotropic meshes. Liu et al. [11] studied a πΆ1 -conforming FE method for nonlinear fourthorder wave equation. In this paper, we consider the following fourth-order partial differential equations of parabolic type: π’π‘ +β β (π (x, π‘) β (β β (π (π‘) βπ’))) = π (x, π‘) , π’ (x, π‘) = Ξπ’ (x, π‘) = 0, π’ (x, 0) = π’0 (x) , (x, π‘) β Ξ© × π½, (x, π‘) β πΞ© × π½, x β Ξ©, (1) 2 Advances in Mathematical Physics where Ξ© β π 2 is a bounded convex polygonal domain with the Lipschitz continuous boundary πΞ© and π½ = (0, π] is the time interval with 0 < π < β. The initial value π’0 (x) and π(x, π‘) are given functions; coefficients π = π(π‘) and π = π(x, π‘) satisfy σ΅¨ σ΅¨σ΅¨ σ΅¨σ΅¨ππ‘ (π‘)σ΅¨σ΅¨σ΅¨ β€ π0 , σ΅¨ σ΅¨ A2 : 0 < π0 β€ π (x, π‘) β€ π1 < +β, σ΅¨σ΅¨σ΅¨ππ‘ (x, π‘)σ΅¨σ΅¨σ΅¨ β€ π0 , A1 : 0 < π0 β€ π (π‘) β€ π1 < +β, (2) for some positive constants π0 , π1 , π0 , π1 , and π0 . Chen proposed the EMFE method for second-order linear/quasilinear elliptic equation [14β16] and fourth-order linear elliptic equation [1]. Then the method was applied to other partial differential equations [17, 18]. Compared to standard MFE methods, the EMFE method is expanded in the sense that three variables are explicitly approximated, namely, the scalar unknown, its gradient, and its flux. In recent years, many numerical method based on the expanded mixed scheme, such as two-grid EMFE methods [19β22], π»1 Galerkin EMFE methods [23β25], expanded mixed covolume method [26], expanded mixed hybrid method [27], and positive definite EMFE method [28], have been proposed and discussed. In [29], we developed and analyzed a new EMFE method for second-order elliptic problems based on the mixed weak formulation [30β32]. The new EMFE method [29], whose gradient belongs to the square integrable (πΏ2 (Ξ©))2 space instead of the classical H(div; Ξ©) space, is different from Chenβs EMFE method [14β16]. In this paper, we present a new coupling method of new EMFE scheme [29] and FE scheme for fourth-order partial differential equation of parabolic type. We first introduce an auxiliary variable πΎ = ββ β (π(π‘)βπ’) to reduce the fourthorder parabolic equation to a coupled system of second-order equations, then solve a second-order equation by FE method. For the other one, we introduce the two auxiliary variables, π = βπ’ and π = βπ(π‘)βπ’ = βππ, to split the equation into a first-order system of equations, then approximate that by a new EMFE method. Because the new EMFE methodβs gradient belongs to the simple square integrable (πΏ2 (Ξ©))2 space, the regularity requirement on the gradient solution π = βπ’ is reduced. We apply the new coupling method of new EMFE and FE to derive a priori error estimates based on both semidiscrete and fully discrete schemes; that is to say, we obtain the optimal a priori error estimates in πΏ2 and π»1 norm for both the scalar unknown π’ and the diffusion term πΎ and a priori error estimates in (πΏ2 )2 -norm for its gradient π and its flux π. Finally, we provide some numerical results to verify our theoretical analysis. Throughout this paper, πΆ will denote a generic positive constant which does not depend on the spatial mesh parameter β or the time step Ξπ‘. At the same time, we denote the natural inner product in πΏ2 (Ξ©) or (πΏ2 (Ξ©))2 by (β , β ) with the corresponding norm β β β. The other notations and definitions for the Sobolev spaces as in [33, 34] are used. 2. New Expanded Mixed Scheme We first introduce an auxiliary variable πΎ = ββ β (π(π‘)βπ’) to obtain the following coupled system of second-order equations: π’π‘ β β β (π (x, π‘) βπΎ) = π (x, π‘) , (3a) πΎ + β β (π (π‘) βπ’) = 0, (3b) π’ (x, π‘) = πΎ (x, π‘) = 0. (3c) For (3b), we introduce the two auxiliary variables, π = βπ’ and π = βπ(π‘)βπ’ = βππ, to obtain the following first-order system of equations: π’π‘ β β β (πβπΎ) = π (x, π‘) , (4a) πΎ β β β π = 0, (4b) π + ππ = 0, (4c) π β βπ’ = 0, (4d) π’ (x, π‘) = πΎ (x, π‘) = 0. (4e) Using Greenβs formula, the new mixed weak formulation of (4a)β(4e) is to determine {π’, πΎ, π, π} : [0, π] σ³¨β π»01 (Ξ©) × π»01 (Ξ©) × (πΏ2 (Ξ©))2 × (πΏ2 (Ξ©))2 such that βV β π»01 (Ξ©) , (π’π‘ , V) + (πβπΎ, βV) = (π, V) , (πΎ, π) + (π, βπ) = 0, βπ β π»01 (Ξ©) , 2 (π, z) + (ππ, z) = 0, βz β (πΏ2 (Ξ©)) , (π, w) β (βπ’, w) = 0, βw β (πΏ2 (Ξ©)) . 2 (5a) (5b) (5c) (5d) Then, the semidiscrete coupled EMFE scheme-FE scheme for (5a)β(5d) is to find {π’β , πΎβ , πβ , πβ } : [0, π] σ³¨β πβ × πβ × Wβ × Wβ such that (π’βπ‘ , Vβ ) + (πβπΎβ , βVβ ) = (π, Vβ ) , βVβ β πβ , (6a) (πΎβ , πβ ) + (πβ , βπβ ) = 0, βπβ β πβ , (6b) (πβ , zβ ) + (ππβ , zβ ) = 0, βzβ β Wβ , (6c) (πβ , wβ ) β (βπ’β , wβ ) = 0, βwβ β Wβ . (6d) By the theory of differential equations [35], we can prove the existence and uniqueness of solution for semidiscrete scheme (6a)β(6d). Remark 1. From the new coupling weak formulation (5a)β (5d), we can find that the gradient belongs to the weaker square integrable (πΏ2 (Ξ©))2 space taking the place of the classical H(div; Ξ©) space. It is easy to see that our method reduces the regularity requirement on the gradient solution π = βπ’. Advances in Mathematical Physics 3 Remark 2. In the semidiscrete coupling scheme (6a)β(6d), the MFE space (πβ , Wβ ) based on the FE pair π1 β π02 is chosen as follows: πβ = {Vβ β πΆ0 (Ξ©) β© π»01 (Ξ©) | Vβ β π1 (πΎ) , βπΎ β Kβ } , 2 π€1β , π€2β β π0 (πΎ) , βπΎ β Kβ } . (7) From [30, 31], we know that MFE space (πβ , Wβ ) satisfies the discrete LBB condition. Remark 3. From (6a)β(6d), we know that (6a) is an FE scheme for π’β , πΎβ , Vβ β πβ and that (6b)β(6d) is a new EMFE scheme for (πΎβ , πβ , πβ ), (πβ , zβ , wβ ) β πβ × Wβ × Wβ . So the scheme (6a)β(6d) is called as the coupling system of new EMFE scheme and FE scheme. Remark 4. The finite element spaces πβ and Wβ for the new EMFE method are chosen as finite dimensional subspaces of π»01 (Ξ©) and (πΏ2 (Ξ©))2 , respectively. However, πβ and Wβ for π»1 -Galerkin MFE method in two space variables in [8] are finite dimensional subspaces of π»01 (Ξ©) and H(div; Ξ©), respectively. Compared to the complex H(div; Ξ©) space used in the π»1 -Galerkin MFE method, we use the weaker (πΏ2 (Ξ©))2 space for our EMFE method. For deriving some a priori error estimates of our method, we first introduce the projection operator Pβ and the new expanded mixed elliptic projection operator Rβ associated with the coupled equations. Lemma 5. One now defines a linear projection operator Pβ : π»01 (Ξ©) β πβ as βVβ β πβ , and then σ΅©σ΅© σ΅© σ΅© σ΅© 2σ΅© σ΅© σ΅©σ΅©πΎ β Pβ πΎσ΅©σ΅©σ΅©πΏ2 (Ξ©) + βσ΅©σ΅©σ΅©πΎ β Pβ πΎσ΅©σ΅©σ΅©π»1 β€ πΆβ σ΅©σ΅©σ΅©πΎσ΅©σ΅©σ΅©π»2 , σ΅© σ΅©σ΅© σ΅© σ΅© 2 σ΅© σ΅© σ΅©σ΅©πΎπ‘ β Pβ πΎπ‘ σ΅©σ΅©σ΅©πΏ2 (Ξ©) β€ πΆβ (σ΅©σ΅©σ΅©πΎσ΅©σ΅©σ΅©π»2 + σ΅©σ΅©σ΅©πΎπ‘ σ΅©σ΅©σ΅©π»2 ) , σ΅© σ΅©σ΅© σ΅© σ΅© σ΅© σ΅© 2 σ΅© σ΅© σ΅©σ΅©πΎπ‘π‘ β Pβ πΎπ‘π‘ σ΅©σ΅©σ΅©πΏ2 (Ξ©) β€ πΆβ (σ΅©σ΅©σ΅©πΎσ΅©σ΅©σ΅©π»2 + σ΅©σ΅©σ΅©πΎπ‘ σ΅©σ΅©σ΅©π»2 + σ΅©σ΅©σ΅©πΎπ‘π‘ σ΅©σ΅©σ΅©π»2 ) . (8) (9) Let (Rβ π’, Rβ π, Rβ π) : [0, π] β πβ × Wβ × Wβ be given by the following new expanded mixed relations [29] βπβ β πβ , Lemma 7. There is a constant πΆ > 0 independent of the spatial mesh parameter β such that σ΅© σ΅©σ΅© 2 σ΅©σ΅©π’ β Rβ π’σ΅©σ΅©σ΅© β€ πΆβ βπ’βπ»2 , σ΅© σ΅©σ΅© 2σ΅© σ΅© σ΅©σ΅©π’π‘ β Rβ π’π‘ σ΅©σ΅©σ΅© β€ πΆβ σ΅©σ΅©σ΅©π’π‘ σ΅©σ΅©σ΅©π»2 , σ΅© σ΅© σ΅©σ΅© 2σ΅© σ΅©σ΅©π’π‘π‘ β Rβ π’π‘π‘ σ΅©σ΅©σ΅© β€ πΆβ σ΅©σ΅©σ΅©π’π‘π‘ σ΅©σ΅©σ΅©π»2 , σ΅©σ΅© σ΅© σ΅©σ΅©π’ β Rβ π’σ΅©σ΅©σ΅©1 β€ πΆβ (βπ’βπ»2 + βπβ(π»1 )2 ) . βzβ β Wβ , (10b) (π β Rβ π, wβ ) β (β (π’ β Rβ π’) , wβ ) = 0, βwβ β Wβ . (10c) (12) In [29], we can obtain the detailed proof for Lemmas 6-7. For a priori error estimates, we write the errors as πΎ β πΎβ = πΎ β Pβ πΎ + Pβ πΎ β πΎβ = π + π; π β πβ = π β Rβ π + Rβ π β πβ = πΏ + π; (13) π β πβ = π β Rβ π + Rβ π β πβ = π + π. Using (5a), (5b), (5c), (5d), (6a), (6b), (6c), (6d), (8), (10a), (10b), and (10c) we can obtain the error equations (ππ‘ , Vβ ) + (πβπ, βVβ ) = β (ππ‘ , Vβ ) , βVβ β πβ , βπβ β πβ , (π, πβ ) + (π, βπβ ) = β (π, πβ ) , (14a) (14b) (π, zβ ) + (ππ, zβ ) = 0, βzβ β Wβ , (14c) (π, wβ ) β (βπ, wβ ) = 0, βwβ β Wβ . (14d) We will prove the error estimates for semidiscrete scheme. Theorem 8. Suppose that πΎ, πΎπ‘ , π’π‘ β πΏ2 (π»2 (Ξ©)), π, π β πΏβ ((π»1 (Ξ©))2 ), and π’, πΎ β πΏβ (π»2 (Ξ©)); then there exists a constant πΆ > 0 independent of the spatial mesh parameter β such that (10a) (π β Rβ π, zβ ) + (π (π β Rβ π) , zβ ) = 0, (11) σ΅© σ΅©σ΅© σ΅©σ΅©β (π’ β Rβ π’)σ΅©σ΅©σ΅© β€ πΆβ (βπ’βπ»2 + βπβ(π»1 )2 ) . π’ β π’β = π’ β Rβ π’ + Rβ π’ β π’β = π + π; 3. Semidiscrete Error Estimates (π β Rβ π, βπβ ) = 0, Lemma 6. There is a constant πΆ > 0 independent of the spatial mesh parameter β such that σ΅© σ΅©σ΅© σ΅©σ΅©π β Rβ πσ΅©σ΅©σ΅© β€ πΆβ (βπβ(π»1 )2 + βπ’βπ»2 ) , σ΅© σ΅©σ΅© σ΅©σ΅©π β Rβ πσ΅©σ΅©σ΅© β€ πΆβ (βπ’βπ»2 + βπβ(π»1 )2 + βπβ(π»1 )2 ) , Wβ = {wβ = (π€1β , π€2β ) β (πΏ2 (Ξ©)) | (πβ (πΎ β Pβ πΎ) , βVβ ) = 0, Then the following two important lemmas based on the new expanded mixed projection (10a)β(10c) hold. σ΅© σ΅© σ΅© σ΅©σ΅© 2 σ΅©σ΅©π’ β π’β σ΅©σ΅©σ΅© β€ πΆβ [βπ’βπΏβ (π»2 ) + σ΅©σ΅©σ΅©πΎσ΅©σ΅©σ΅©πΏβ (π»2 ) π‘ +(β« |||βΌ|||2 ππ ) 0 1/2 ], 4 Advances in Mathematical Physics σ΅© σ΅©σ΅© σ΅©σ΅©π’ β π’β σ΅©σ΅©σ΅©1 β€ πΆβ [βπβπΏβ ((π»1 )2 ) + βπ’βπΏβ (π»2 ) Using (18), (19), Cauchy-Schwarz inequality and Young inequality, we have π‘ 1/2 π‘ σ΅© σ΅© +σ΅©σ΅©σ΅©πΎσ΅©σ΅©σ΅©πΏβ (π»2 ) + (β« |||βΌ|||2 ππ ) 0 π0 ββπβ2 + 2π0 β« ββπβ2 ππ ], 0 π‘ σ΅© σ΅© σ΅© σ΅©σ΅© σ΅©σ΅©π β πβ σ΅©σ΅©σ΅© β€ πΆβ [βπβπΏβ ((π»1 )2 ) + σ΅©σ΅©σ΅©πΎσ΅©σ΅©σ΅©πΏβ (π»2 ) π‘ σ΅© σ΅© σ΅© σ΅© β€ 2πΆ1 σ΅©σ΅©σ΅©πσ΅©σ΅©σ΅© ββπβ + 2πΆ2 β« σ΅©σ΅©σ΅©ππ‘ σ΅©σ΅©σ΅© ββπβ ππ 0 1/2 π‘ +(β« |||βΌ|||2 ππ ) ], 0 π‘ σ΅© σ΅©2 + πΆ β« (πΆ12 ββπβ2 + σ΅©σ΅©σ΅©ππ‘ σ΅©σ΅©σ΅© ) ππ σ΅© σ΅©σ΅© σ΅©σ΅©π β πβ σ΅©σ΅©σ΅© β€ πΆβ [βπ’βπΏβ (π»2 ) + βπβπΏβ ((π»1 )2 ) + βπβπΏβ ((π»1 )2 ) 1/2 π‘ σ΅© σ΅© +σ΅©σ΅©σ΅©πΎσ΅©σ΅©σ΅©πΏβ (π»2 ) + (β« |||βΌ|||2 ππ ) 0 σ΅© σ΅©2 (ππ‘ , π) + σ΅©σ΅©σ΅©σ΅©π1/2 βπσ΅©σ΅©σ΅©σ΅© = β (ππ‘ , π) , (16a) β (π, ππ‘ ) β (π, βππ‘ ) = (π, ππ‘ ) , (16b) (π, βππ‘ ) + (ππ, βππ‘ ) = 0, (16c) 1 π σ΅©σ΅© 1/2 σ΅©σ΅©2 1 σ΅©σ΅©π βπσ΅©σ΅© = (ππ‘ βπ, βπ) . σ΅© 2 ππ‘ σ΅© 2 (16d) Adding the above four equations and using Cauchy-Schwarz inequality and Youngβs inequality, we have 0 π‘ σ΅© σ΅©2 + πΆ β« (πΆ12 ββπβ2 + σ΅©σ΅©σ΅©ππ‘ σ΅©σ΅©σ΅© ) ππ . (20) Using Gronwallβs lemma, we get π‘ ββπβ2 + β« ββπβ2 ππ 0 π‘ σ΅© σ΅©2 σ΅© σ΅©2 σ΅© σ΅© 2 σ΅© σ΅©2 β€ πΆ (σ΅©σ΅©σ΅©πσ΅©σ΅©σ΅© + β« (σ΅©σ΅©σ΅©ππ‘ σ΅©σ΅©σ΅© + σ΅©σ΅©σ΅©πσ΅©σ΅©σ΅© + σ΅©σ΅©σ΅©ππ‘ σ΅©σ΅©σ΅© ) ππ ) . 0 π‘ βπβ2 + β« βπβ2 ππ 0 π‘ σ΅©2 1 (π βπ, βπ) 2 π‘ π 1 = β (ππ‘ , π) + (π, π) β (ππ‘ , π) + (ππ‘ βπ, βπ) ππ‘ 2 (21) Substitute (19) into (21) to have σ΅© σ΅©2 σ΅© σ΅©2 σ΅© σ΅© 2 σ΅© σ΅©2 β€ πΆ (σ΅©σ΅©σ΅©πσ΅©σ΅©σ΅© + β« (σ΅©σ΅©σ΅©ππ‘ σ΅©σ΅©σ΅© + σ΅©σ΅©σ΅©πσ΅©σ΅©σ΅© + σ΅©σ΅©σ΅©ππ‘ σ΅©σ΅©σ΅© ) ππ ) . 0 1 π σ΅©σ΅© 1/2 σ΅© σ΅©σ΅© 1/2 σ΅© σ΅©σ΅©π βπσ΅©σ΅© + σ΅©σ΅©π βπσ΅©σ΅© σ΅© σ΅© σ΅© 2 ππ‘ σ΅© β€ 4πΆ2 π‘ σ΅© σ΅©2 σ΅©2 π 2 σ΅©σ΅©πσ΅©σ΅©σ΅© + 0 ββπβ + 2 β« σ΅©σ΅©σ΅©ππ‘ σ΅©σ΅©σ΅© ππ π0 2 π0 0 0 Proof. Choose Vβ = π, πβ = ππ‘ , zβ = βππ‘ , and wβ = βππ‘ in (14a)β(14d) to obtain = β (ππ‘ , π) + (π, ππ‘ ) + β€ + π0 β« ββπβ2 ππ where |||βΌ|||2 β βπ’π‘ β2π»2 + βπΎβ2π»2 + βπΎπ‘ β2π»2 . σ΅©2 0 8πΆ12 σ΅©σ΅© π‘ ], (15) β (ππ, βππ‘ ) + π‘ σ΅© σ΅© σ΅© σ΅©2 σ΅© σ΅© β€ 2 σ΅©σ΅©σ΅©πσ΅©σ΅©σ΅© βπβ + 2 β« σ΅©σ΅©σ΅©ππ‘ σ΅©σ΅©σ΅© βπβ ππ + πΆ β« (βπβ21 + σ΅©σ΅©σ΅©ππ‘ σ΅©σ΅©σ΅© ) ππ 0 0 (22) Taking wβ = π in (14d) and using (22), we get (17) π σ΅© σ΅©2 σ΅© σ΅© (π, π) + σ΅©σ΅©σ΅©ππ‘ σ΅©σ΅©σ΅© βπβ + πΆ (βπβ21 + σ΅©σ΅©σ΅©ππ‘ σ΅©σ΅©σ΅© ) . ππ‘ π‘ σ΅© σ΅©2 σ΅© σ΅©2 σ΅© σ΅© 2 σ΅© σ΅©2 βπβ2 β€ πΆ (σ΅©σ΅©σ΅©πσ΅©σ΅©σ΅© + β« (σ΅©σ΅©σ΅©ππ‘ σ΅©σ΅©σ΅© + σ΅©σ΅©σ΅©πσ΅©σ΅©σ΅© + σ΅©σ΅©σ΅©ππ‘ σ΅©σ΅©σ΅© ) ππ ) . 0 (23) Choose zβ = π in (14c) and use (22) to obtain π‘ σ΅© σ΅©2 σ΅© σ΅©2 σ΅© σ΅© 2 σ΅© σ΅©2 βπβ2 β€ πΆ (σ΅©σ΅©σ΅©πσ΅©σ΅©σ΅© + β« (σ΅©σ΅©σ΅©ππ‘ σ΅©σ΅©σ΅© + σ΅©σ΅©σ΅©πσ΅©σ΅©σ΅© + σ΅©σ΅©σ΅©ππ‘ σ΅©σ΅©σ΅© ) ππ ) . 0 Integrate with respect to time from 0 to π‘ to obtain (24) π‘ π0 ββπβ2 + 2π0 β« ββπβ2 ππ Combining Lemmas 5β7, (21)β(24) and using the triangle inequality, we get the error estimates for Theorem 8. 0 π‘ π‘ σ΅© σ΅© σ΅© σ΅©2 β€ 2 (π, π) + 2 β« σ΅©σ΅©σ΅©ππ‘ σ΅©σ΅©σ΅© βπβ ππ + πΆ β« (βπβ21 + σ΅©σ΅©σ΅©ππ‘ σ΅©σ΅©σ΅© ) ππ . 0 0 (18) Noting that π, π β π»01 (Ξ©), we use PoincareΜ inequality to have βπβ1 β€ πΆ1 ββπβ , βπβ β€ πΆ2 ββπβ . (19) Remark 9. From Theorem 8, we can see that σ΅© σ΅© σ΅©σ΅© σ΅© 2 σ΅©σ΅©π’ β π’β σ΅©σ΅©σ΅©πΏβ (πΏ2 ) + βσ΅©σ΅©σ΅©π’ β π’β σ΅©σ΅©σ΅©πΏβ (π»1 ) = π (β ) ; (25) that is to say, a priori error estimates in πΏ2 and π»1 -norms for the scalar unknown π’ are optimal. Advances in Mathematical Physics 5 Theorem 10. Suppose that ππ π’/ππ‘π , ππ πΎ/ππ‘π β πΏ2 (π»2 (Ξ©)) and πΎ, πΎπ‘ , π’π‘ β πΏβ (π»2 (Ξ©)); then there exists a constant πΆ > 0 independent of the spatial mesh parameter β such that σ΅© σ΅© σ΅© σ΅©σ΅© σ΅©σ΅©πΎ β πΎβ σ΅©σ΅©σ΅© + βσ΅©σ΅©σ΅©πΎ β πΎβ σ΅©σ΅©σ΅©1 π‘ σ΅© σ΅©2 π0 ββπβ2 + 2π0 β« σ΅©σ΅©σ΅©βππ‘ σ΅©σ΅©σ΅© ππ 0 π‘ σ΅© σ΅© β€ β2 (ππ‘ , π) + 2πΆ2 β« σ΅©σ΅©σ΅©ππ‘π‘ σ΅©σ΅©σ΅© ββπβ ππ + 2 (ππ‘ , π) 0 σ΅© σ΅© σ΅© σ΅© σ΅© σ΅© β€ πΆβ2 [σ΅©σ΅©σ΅©π’π‘ σ΅©σ΅©σ΅©πΏβ (π»2 ) + σ΅©σ΅©σ΅©πΎσ΅©σ΅©σ΅©πΏβ (π»2 ) + σ΅©σ΅©σ΅©πΎπ‘ σ΅©σ΅©σ΅©πΏβ (π»2 ) [ π‘ 1/2 σ΅©σ΅© π σ΅©σ΅©2 σ΅©σ΅© π σ΅©σ΅©2 σ΅©σ΅© π πΎ σ΅©σ΅© σ΅©σ΅© π π’ σ΅©σ΅© σ΅© σ΅© σ΅© σ΅© + (β« ( βσ΅©σ΅© π σ΅©σ΅© + σ΅©σ΅© π σ΅©σ΅© ) ππ ) ] . 0 σ΅© ππ‘ σ΅©σ΅©π»2 σ΅©σ΅© ππ‘ σ΅©σ΅©π»2 π=0σ΅© ] (26) π‘ Integrate with respect to time from 0 to π‘ and use (19), Cauchy-Schwarz inequality, and Young inequality to get 2 Proof. Differentiating (14b), (14c), and (14d) with respect to time π‘, respectively, we get (ππ‘ , Vβ ) + (πβπ, βVβ ) = β (ππ‘ , Vβ ) , (27a) (ππ‘ , πβ ) + (ππ‘ , βπβ ) = β (ππ‘ , πβ ) , (27b) (ππ‘ , zβ ) + (ππ‘ π, zβ ) + (πππ‘ , zβ ) = 0, (27c) (ππ‘ , wβ ) β (βππ‘ , wβ ) = 0. (27d) Choose Vβ = ππ‘ , πβ = ππ‘ , zβ = βππ‘ , and wβ = βππ‘ in (27a)β (27d) to get 1 π σ΅©σ΅© 1/2 σ΅©σ΅©2 1 σ΅©σ΅©π βπσ΅©σ΅© = β (ππ‘ , ππ‘ ) + (ππ‘ βπ, βπ) , (ππ‘ , ππ‘ ) + σ΅© σ΅© 2 ππ‘ 2 (28a) β (ππ‘ , ππ‘ ) β (ππ‘ , βππ‘ ) = (ππ‘ , ππ‘ ) , (28b) (ππ‘ , βππ‘ ) + (ππ‘ π, βππ‘ ) + (πππ‘ , βππ‘ ) = 0, (28c) σ΅©2 σ΅© β (πππ‘ , βππ‘ ) + σ΅©σ΅©σ΅©σ΅©π1/2 βππ‘ σ΅©σ΅©σ΅©σ΅© = 0. (28d) π‘ σ΅© σ΅©2 σ΅© σ΅©2 + πΆ β« (σ΅©σ΅©σ΅©ππ‘π‘ σ΅©σ΅©σ΅© + βπβ2 + βπβ2 ) ππ + π0 β« σ΅©σ΅©σ΅©βππ‘ σ΅©σ΅©σ΅© ππ 0 0 β€ π‘ 8πΆ22 σ΅©σ΅© σ΅©σ΅©2 π0 σ΅© σ΅©2 σ΅© σ΅©2 2 2 σ΅©σ΅©ππ‘ σ΅©σ΅© + ββπβ + σ΅©σ΅©σ΅©ππ‘ σ΅©σ΅©σ΅© + βπβ + π0 β« σ΅©σ΅©σ΅©βππ‘ σ΅©σ΅©σ΅© ππ π0 2 0 π‘ σ΅© σ΅©2 σ΅© σ΅©2 + πΆ β« (σ΅©σ΅©σ΅©ππ‘π‘ σ΅©σ΅©σ΅© + βπβ2 + βπβ2 + σ΅©σ΅©σ΅©ππ‘π‘ σ΅©σ΅©σ΅© + ββπβ2 ) ππ . 0 (30) Substituting (22) and (23) into (30) and using Gronwall lemma, we get π‘ σ΅© σ΅©2 ββπβ2 + β« σ΅©σ΅©σ΅©βππ‘ σ΅©σ΅©σ΅© ππ 0 σ΅© σ΅©2 σ΅© σ΅©2 σ΅© σ΅©2 β€ πΆσ΅©σ΅©σ΅©ππ‘ σ΅©σ΅©σ΅© + σ΅©σ΅©σ΅©πσ΅©σ΅©σ΅© + σ΅©σ΅©σ΅©ππ‘ σ΅©σ΅©σ΅© π‘ σ΅© σ΅©2 σ΅© σ΅©2 σ΅© σ΅©2 + πΆ β« (σ΅©σ΅©σ΅©πσ΅©σ΅©σ΅© + σ΅©σ΅©σ΅©ππ‘ σ΅©σ΅©σ΅© + σ΅©σ΅©σ΅©ππ‘π‘ σ΅©σ΅©σ΅© 0 (31) σ΅© σ΅©2 σ΅© σ΅©2 σ΅© σ΅©2 +σ΅©σ΅©σ΅©πσ΅©σ΅©σ΅© + σ΅©σ΅©σ΅©ππ‘ σ΅©σ΅©σ΅© + σ΅©σ΅©σ΅©ππ‘π‘ σ΅©σ΅©σ΅© ) ππ . Use (19) to obtain σ΅© σ΅©2 σ΅© σ΅©2 σ΅© σ΅©2 βπβ2 β€ πΆσ΅©σ΅©σ΅©ππ‘ σ΅©σ΅©σ΅© + σ΅©σ΅©σ΅©πσ΅©σ΅©σ΅© + σ΅©σ΅©σ΅©ππ‘ σ΅©σ΅©σ΅© π‘ σ΅© σ΅©2 σ΅© σ΅©2 σ΅© σ΅©2 + πΆ β« (σ΅©σ΅©σ΅©πσ΅©σ΅©σ΅© + σ΅©σ΅©σ΅©ππ‘ σ΅©σ΅©σ΅© + σ΅©σ΅©σ΅©ππ‘π‘ σ΅©σ΅©σ΅© 0 (32) σ΅© σ΅© 2 σ΅© σ΅©2 σ΅© σ΅©2 + σ΅©σ΅©σ΅©πσ΅©σ΅©σ΅© + σ΅©σ΅©σ΅©ππ‘ σ΅©σ΅©σ΅© + σ΅©σ΅©σ΅©ππ‘π‘ σ΅©σ΅©σ΅© ) ππ . Combining Lemmas 5β7, (31), and (32) and using the triangle inequality, we obtain the error estimates for Theorem 10. Remark 11. From Theorem 10, we can see that σ΅©σ΅© σ΅© σ΅© σ΅© 2 σ΅©σ΅©πΎ β πΎβ σ΅©σ΅©σ΅©πΏβ (πΏ2 ) + βσ΅©σ΅©σ΅©πΎ β πΎβ σ΅©σ΅©σ΅©πΏβ (π»1 ) = π (β ) ; Adding the four equations for (28a)β(28d), we have that is to say, a priori error estimates in πΏ2 and π»1 -norm for the diffusion term πΎ are optimal. 1 π σ΅©σ΅© 1/2 σ΅©σ΅©2 σ΅©σ΅© 1/2 σ΅©σ΅©2 σ΅©σ΅©π βπσ΅©σ΅© + σ΅©σ΅©π βππ‘ σ΅©σ΅© σ΅© σ΅© σ΅© 2 ππ‘ σ΅© = β (ππ‘ , ππ‘ ) + (ππ‘ , ππ‘ ) β (ππ‘ π, βππ‘ ) + =β 1 (π βπ, βπ) 2 π‘ π π (π , π) + (ππ‘π‘ , π) + (π , π) ππ‘ π‘ ππ‘ π‘ β (ππ‘π‘ , π) β (ππ‘ π, βππ‘ ) + (33) 1 (π βπ, βπ) . 2 π‘ (29) 4. Fully Discrete Error Estimates In the following analysis, we will derive a priori error estimates based on fully discrete backward Euler scheme. Let 0 = π‘0 < π‘1 < π‘2 < β β β < π‘π = π be a given partition of the time interval [0, π] with step length Ξπ‘ = π/π and nodes π‘π = πΞπ‘, for some positive integer π. For a smooth function π on [0, π], define ππ = π(π‘π ). 6 Advances in Mathematical Physics For (5a), (5b), (5c), and (5d), we have the following equivalent formulation: ( π’π β π’πβ1 , V)+(ππ βπΎπ , βV) = (ππ +π 1π , V) , Ξπ‘ βV β π»01 (Ξ©) , (34a) (πΎπ , π) + (ππ , βπ) = 0, Proof. Using (8), (10a), (10b), (10c), (34a), (34b), (34c), (34d), (36a), (36b), (36c), and (36d) at π‘ = π‘π , we obtain the error equations βπ β π»01 (Ξ©) , (34b) 2 (ππ , z) + (ππ ππ , z) = 0, βz β (πΏ2 (Ξ©)) , (ππ , w) β (βπ’π , w) = 0, βw β (πΏ2 (Ξ©)) , (34c) 2 (34d) ( ππ β ππβ1 , Vβ ) + (ππ βππ , βVβ ) Ξπ‘ ππ β ππβ1 = β( , Vβ ) + (π 1π , Vβ ) , Ξπ‘ (ππ , πβ ) + (ππ , βπβ ) = β (ππ , πβ ) , π (π , zβ ) + (π π , zβ ) = 0, where π 1π = π π’ βπ’ Ξπ‘ πβ1 β π’π‘ (π‘π ) = π‘π 1 β« (π‘ β π ) π’π‘π‘ ππ . Ξπ‘ π‘πβ1 πβ1 (35) Now a fully discrete procedure is to find (π’βπ , πΎβπ , ππβ , ππβ ) β πβ × πβ × Wβ × Wβ (π = 0, 1, . . . , π) such that ( π π π’βπ β π’βπβ1 , Vβ ) + (ππ βπΎβπ , βVβ ) = (ππ , Vβ ) , Ξπ‘ βVβ β πβ , (36a) (πΎβπ , πβ ) + (ππβ , βπβ ) = 0, βπβ β πβ , (36b) (ππβ , zβ ) + (ππ ππβ , zβ ) = 0, βzβ β Wβ , (36c) (ππβ , wβ ) β (βπ’βπ , wβ ) = 0, βwβ β Wβ . πΎπ β πΎβπ = πΎπ β Pβ πΎπ + Pβ πΎπ β πΎβπ = ππ + ππ ; π’π β π’βπ = π’π β Rβ π’π + Rβ π’π β π’βπ = ππ + ππ ; π β ππβ π π π = π β Rβ π + Rβ π β ππβ π π =πΏ +π ; ππ β ππβ = ππ β Rβ ππ + Rβ ππ β ππβ = ππ + ππ . We will prove the theorem for the fully discrete error estimates. Theorem 12. Suppose that πΎ, πΎπ‘ , π’π‘ β πΏ2 (π»2 (Ξ©)), π, π β πΏβ ((π»1 (Ξ©))2 ), πΎ β πΏβ (π»2 (Ξ©)), and π’π‘π‘ β πΏ2 (πΏ2 (Ξ©)); then there exists a constant πΆ > 0 independent of the spatial mesh parameter β and the time step Ξπ‘ such that σ΅© σ΅©σ΅© π½ σ΅©σ΅©π’ β π’βπ½ σ΅©σ΅©σ΅© β€ πΆβ2 (βπ’βπΏβ (π»2 ) + |||⧫|||) + πΆΞπ‘σ΅©σ΅©σ΅©σ΅©π’π‘π‘ σ΅©σ΅©σ΅©σ΅©πΏ2 (πΏ2 ) , σ΅© σ΅© σ΅©σ΅© π½ π½σ΅© σ΅©σ΅©π’ β π’β σ΅©σ΅©σ΅© β€ πΆβ (βπ’βπΏβ (π»2 ) + βπβπΏβ ((π»1 )2 ) + |||⧫|||) σ΅©1 σ΅© σ΅© σ΅© + πΆΞπ‘σ΅©σ΅©σ΅©π’π‘π‘ σ΅©σ΅©σ΅©πΏ2 (πΏ2 ) , σ΅© σ΅©σ΅© π½ σ΅©σ΅©π β ππ½β σ΅©σ΅©σ΅© β€ πΆβ (βπβπΏβ ((π»1 )2 ) + |||⧫|||) + πΆΞπ‘σ΅©σ΅©σ΅©σ΅©π’π‘π‘ σ΅©σ΅©σ΅©σ΅©πΏ2 (πΏ2 ) , σ΅© σ΅© σ΅©σ΅© π½ π½σ΅© σ΅©σ΅©π β πβ σ΅©σ΅©σ΅© β€ πΆβ (βπβπΏβ ((π»1 )2 ) + |||⧫|||) + πΆΞπ‘σ΅©σ΅©σ΅©σ΅©π’π‘π‘ σ΅©σ΅©σ΅©σ΅©πΏ2 (πΏ2 ) , σ΅© σ΅© (38) where |||⧫||| β βπ’π‘ βπΏ2 (π»2 ) + βπΎβπΏβ (π»2 ) + βπΎβπΏ2 (π»2 ) + βπΎπ‘ βπΏ2 (π»2 ) . βπβ β πβ , βzβ β Wβ , βwβ β Wβ . (39b) (39c) (39d) σ΅©σ΅© π 1/2 π σ΅©σ΅©2 σ΅©σ΅©(π ) βπ σ΅©σ΅© σ΅©σ΅© σ΅©σ΅© = β( ππ β ππβ1 π ππ β ππβ1 π ,π ) β ( , π ) + (π 1π , ππ ) Ξπ‘ Ξπ‘ = (ππ , ππ β ππβ1 π βππ β βππβ1 )β( , π ) + (π 1π , ππ ) Ξπ‘ Ξπ‘ + (ππ , (37) βVβ β πβ , Take Vβ = ππ in (39a), πβ = (ππ β ππβ1 )/Ξπ‘ in (39b), zβ = (βππ β βππβ1 )/Ξπ‘ in (39c), and wβ = (βππ β βππβ1 )/Ξπ‘ in (39d) to get (36d) For deriving the fully discrete error estimates, we now decompose the errors as π (ππ , wβ ) β (βππ , wβ ) = 0, (39a) ππ β ππβ1 ) Ξπ‘ = β (ππ ππ , ππ β ππβ1 π βππ β βππβ1 )β( ,π ) Ξπ‘ Ξπ‘ + (π 1π , ππ ) + (ππ , = β (ππ ππ β ππβ1 π βππ β βππβ1 , βππ ) β ( ,π ) Ξπ‘ Ξπ‘ + (π 1π , ππ ) + (ππ , =β ππ β ππβ1 ) Ξπ‘ ππ β ππβ1 ) Ξπ‘ 1 σ΅©σ΅©σ΅© π 1/2 π σ΅©σ΅©σ΅©2 σ΅©σ΅©σ΅© πβ1 1/2 πβ1 σ΅©σ΅©σ΅©2 (σ΅©(π ) βπ σ΅©σ΅© β σ΅©σ΅©σ΅©(π ) βπ σ΅©σ΅©σ΅© ) σ΅© σ΅© 2Ξπ‘ σ΅©σ΅© σ΅© β 1 σ΅©σ΅© π σ΅©2 σ΅©σ΅©π (βππ β βππβ1 )σ΅©σ΅©σ΅© σ΅© 2Ξπ‘ σ΅© + ππ β ππβ1 π 1 ππ β ππβ1 πβ1 πβ1 ( βπ , βπ ) β ( ,π ) 2 Ξπ‘ Ξπ‘ + (π 1π , ππ ) + (ππ , ππ β ππβ1 ). Ξπ‘ (40) Advances in Mathematical Physics 7 By (40), we have Use (42) and Gronwall lemma to get π½ σ΅©σ΅© π½ σ΅©σ΅©2 σ΅©σ΅©βπ σ΅©σ΅© + Ξπ‘ β σ΅©σ΅©σ΅©σ΅©βππ σ΅©σ΅©σ΅©σ΅©2 σ΅© σ΅© σ΅©σ΅© π 1/2 π σ΅©σ΅©2 σ΅©σ΅©(π ) βπ σ΅©σ΅© + 1 σ΅©σ΅©σ΅©σ΅©ππ (βππ β βππβ1 )σ΅©σ΅©σ΅©σ΅©2 σ΅©σ΅© σ΅©σ΅© σ΅© 2Ξπ‘ σ΅© π=1 σ΅© σ΅©2 β€ πΆσ΅©σ΅©σ΅©σ΅©ππ½ σ΅©σ΅©σ΅©σ΅© 1 σ΅©σ΅©σ΅© π 1/2 π σ΅©σ΅©σ΅©2 σ΅©σ΅©σ΅© πβ1 1/2 πβ1 σ΅©σ΅©σ΅©2 + (σ΅©(π ) βπ σ΅©σ΅© β σ΅©σ΅©σ΅©(π ) βπ σ΅©σ΅©σ΅© ) σ΅© σ΅© 2Ξπ‘ σ΅©σ΅© σ΅© = σ΅©σ΅© π σ΅©2 σ΅© σ΅©2 σ΅©σ΅© π β ππβ1 σ΅©σ΅©σ΅© σ΅©σ΅©σ΅© ππ β ππβ1 σ΅©σ΅©σ΅© σ΅©σ΅© π σ΅©σ΅©2 σ΅© σ΅© σ΅© σ΅©σ΅© + σ΅©σ΅©π 1 σ΅©σ΅© ) . + πΆΞπ‘ β (σ΅©σ΅© σ΅©σ΅© + σ΅©σ΅© σ΅©σ΅© Ξπ‘ σ΅© Ξπ‘ σ΅©σ΅© σ΅©σ΅© σ΅© π=1 σ΅© (43) π½ ππ β ππβ1 π 1 ππ β ππβ1 πβ1 πβ1 ( βπ , βπ ) β ( ,π ) 2 Ξπ‘ Ξπ‘ Note that ππ β ππβ1 + (π 1π , ππ ) + (ππ , ) Ξπ‘ = π‘ π σ΅© σ΅©2 σ΅©σ΅© π σ΅©σ΅©2 σ΅©σ΅©π 1 σ΅©σ΅© β€ πΆΞπ‘ β« σ΅©σ΅©σ΅©π’π‘π‘ σ΅©σ΅©σ΅© ππ , π‘ πβ1 ππ β ππβ1 π 1 1 π‘π ( β« ππ‘ (π ) ππ βππβ1 , βππβ1 ) β ( ,π ) 2 Ξπ‘ π‘πβ1 Ξπ‘ + (π 1π , ππ ) + β (ππβ1 , (ππ , ππ ) β (ππβ1 , ππβ1 ) Ξπ‘ π π βπ Ξπ‘ πβ1 ). Multiplying by 2Ξπ‘, summing (41) from π = 1 to π½, and using (19), the resulting equation becomes π½ σ΅©σ΅© π½ σ΅©σ΅©2 σ΅©σ΅©βπ σ΅©σ΅© + Ξπ‘ β σ΅©σ΅©σ΅©σ΅©βππ σ΅©σ΅©σ΅©σ΅©2 σ΅© σ΅© π=1 π‘π½ σ΅© σ΅©2 σ΅©2 σ΅© σ΅©2 σ΅© β€ πΆ (σ΅©σ΅©σ΅©σ΅©ππ½ σ΅©σ΅©σ΅©σ΅© + β« (σ΅©σ΅©σ΅©ππ‘ (π )σ΅©σ΅©σ΅© + σ΅©σ΅©σ΅©ππ‘ (π )σ΅©σ΅©σ΅© ) ππ π‘ π½ σ΅© σ΅©2 + (Ξπ‘)2 β« σ΅©σ΅©σ΅©π’π‘π‘ σ΅©σ΅©σ΅© ππ ) . π‘ Combining (19) and (45), we obtain π=1 π½ σ΅©σ΅© π½ σ΅©σ΅©2 σ΅©σ΅©π σ΅©σ΅© + Ξπ‘ β σ΅©σ΅©σ΅©σ΅©ππ σ΅©σ΅©σ΅©σ΅©2 σ΅© σ΅© σ΅©σ΅© π σ΅© π½ σ΅© π β ππβ1 σ΅©σ΅©σ΅© σ΅©σ΅© π σ΅©σ΅© σ΅©σ΅© π σ΅©σ΅© σ΅©σ΅© + σ΅©σ΅©π 1 σ΅©σ΅©) σ΅©σ΅©π σ΅©σ΅© β€ 2Ξπ‘ β (σ΅©σ΅©σ΅©σ΅© σ΅© σ΅© Ξπ‘ σ΅©σ΅© π=1 σ΅© π=1 π‘π½ σ΅© σ΅©2 σ΅©2 σ΅© σ΅©2 σ΅© β€ πΆ (σ΅©σ΅©σ΅©σ΅©ππ½ σ΅©σ΅©σ΅©σ΅© + β« (σ΅©σ΅©σ΅©ππ‘ (π )σ΅©σ΅©σ΅© + σ΅©σ΅©σ΅©ππ‘ (π )σ΅©σ΅©σ΅© ) ππ (46) π‘0 π½β1 σ΅¨ σ΅¨ σ΅© σ΅©2 sup σ΅¨σ΅¨σ΅¨ππ‘ (π‘)σ΅¨σ΅¨σ΅¨ Ξπ‘ β σ΅©σ΅©σ΅©βππ σ΅©σ΅©σ΅© π‘β[π‘πβ1 ,π‘π ] π=1 π‘ π½ σ΅© σ΅©2 + (Ξπ‘)2 β« σ΅©σ΅©σ΅©π’π‘π‘ σ΅©σ΅©σ΅© ππ ) . π‘ 0 Taking wβ = π in (39d) and zβ = ππ in (39c) and using (45), we have σ΅©σ΅©σ΅©ππ½ σ΅©σ΅©σ΅©2 + σ΅©σ΅©σ΅©ππ½ σ΅©σ΅©σ΅©2 σ΅©σ΅© σ΅©σ΅© σ΅©σ΅© σ΅©σ΅© π σ΅© σ΅©2 σ΅©σ΅© πβ1 σ΅©σ΅©2 σ΅©σ΅©σ΅© ππ β ππβ1 σ΅©σ΅©σ΅© σ΅© σ΅©σ΅© σ΅© σ΅© σ΅©σ΅© ) + σ΅©σ΅©σ΅©ππ½ σ΅©σ΅©σ΅© σ΅©σ΅©σ΅©ππ½ σ΅©σ΅©σ΅© σ΅© σ΅© + Ξπ‘ β (σ΅©σ΅©π σ΅©σ΅© + σ΅©σ΅© σ΅© σ΅© σ΅©σ΅© σ΅© Ξπ‘ σ΅©σ΅© σ΅©σ΅© π=1 π½ σ΅©σ΅© π σ΅©2 π½ π½ σ΅© π β ππβ1 σ΅©σ΅©σ΅© σ΅©σ΅© π σ΅©σ΅©2 σ΅©σ΅© + σ΅©σ΅©π 1 σ΅©σ΅© ) + π0 Ξπ‘ β σ΅©σ΅©σ΅©σ΅©βππ σ΅©σ΅©σ΅©σ΅©2 β€ πΆΞπ‘ β (σ΅©σ΅©σ΅©σ΅© σ΅© σ΅© Ξπ‘ σ΅©σ΅© π=1 σ΅© π=1 π‘ π½ σ΅© σ΅©2 σ΅©2 σ΅© σ΅©2 σ΅© β€ πΆ (σ΅©σ΅©σ΅©σ΅©ππ½ σ΅©σ΅©σ΅©σ΅© + β« (σ΅©σ΅©σ΅©ππ‘ (π )σ΅©σ΅©σ΅© + σ΅©σ΅©σ΅©ππ‘ (π )σ΅©σ΅©σ΅© ) ππ π‘ 0 (47) π‘ π½ σ΅© σ΅©2 +(Ξπ‘)2 β« σ΅©σ΅©σ΅©π’π‘π‘ σ΅©σ΅©σ΅© ππ ) . π‘ σ΅© σ΅©2 σ΅©σ΅© πβ1 σ΅©σ΅©2 σ΅©σ΅©σ΅© ππ β ππβ1 σ΅©σ΅©σ΅© σ΅© σ΅©2 σ΅©σ΅© ) + πΆσ΅©σ΅©σ΅©ππ½ σ΅©σ΅©σ΅© + πΆΞπ‘ β (σ΅©σ΅©σ΅©βπ σ΅©σ΅©σ΅© + σ΅©σ΅©σ΅© σ΅© σ΅© σ΅© Ξπ‘ σ΅©σ΅© σ΅©σ΅© π=1 π½ π0 σ΅©σ΅© π½ σ΅©σ΅©2 σ΅©σ΅©βπ σ΅©σ΅© . 2σ΅© σ΅© (45) π‘0 0 π½ σ΅© σ΅©2 σ΅© σ΅©2 π0 σ΅©σ΅©σ΅©σ΅©βππ½ σ΅©σ΅©σ΅©σ΅© + 2π0 Ξπ‘ β σ΅©σ΅©σ΅©βππ σ΅©σ΅©σ΅© + (44) Substitute (44) into (43) to get (41) + σ΅©σ΅© π 1 π‘π σ΅©σ΅© σ΅©σ΅© π β π σ΅©σ΅©σ΅© σ΅©2 σ΅©σ΅© σ΅©σ΅© β€ β« σ΅©σ΅©ππ‘ (π )σ΅©σ΅©σ΅© ππ , σ΅©σ΅© Ξπ‘ σ΅©σ΅© Ξπ‘ π‘πβ1 σ΅© σ΅© 2 σ΅©σ΅© π σ΅© 1 π‘π σ΅©σ΅© σ΅©σ΅© π β ππβ1 σ΅©σ΅©σ΅© σ΅©2 σ΅©σ΅© σ΅©σ΅© β€ β« σ΅©π (π )σ΅©σ΅© ππ . Ξπ‘ Ξπ‘ π‘πβ1 σ΅© π‘ σ΅© σ΅©σ΅©σ΅© σ΅©σ΅©σ΅© πβ1 σ΅©2 0 Combining Lemmas 5β7, (45), (46), and (47) with the triangle inequality, we accomplish the proof for Theorem 12. (42) In the following analysis, we will derive the optimal a priori error estimates in πΏ2 and π»1 -norm for πΎ. 8 Advances in Mathematical Physics Table 1: πΏ2 and π»1 -norms and convergence order for π’. (β, Ξπ‘) βπ’ β π’β βπΏβ (πΏ2 (Ξ©)) βπ’ β π’β βπΏβ (π»1 (Ξ©)) Order Order (β2/8, 1/8) 2.8933π β 002 (β2/16, 1/16) 8.2158π β 003 1.8162 1.2986π β 001 0.8719 (β2/32, 1/32) 2.1012π β 003 1.9672 6.8466π β 002 0.9235 (β2/64, 1/64) 5.3511π β 004 1.9733 3.5245π β 002 0.9580 2.3766π β 001 Table 2: πΏ2 and π»1 -norms and convergence order for πΎ. (β, Ξπ‘) βπΎ β πΎβ βπΏβ (πΏ2 (Ξ©)) βπΎ β πΎβ βπΏβ (π»1 (Ξ©)) Order Order (β2/8, 1/8) 3.4238π β 001 (β2/16, 1/16) 9.2701π β 002 1.8849 2.5429π + 000 0.8381 (β2/32, 1/32) 2.3840π β 002 1.9592 1.3493π + 000 0.9143 (β2/64, 1/64) 5.8649π β 003 2.0232 6.9558π β 001 0.9559 4.5457π + 000 Theorem 13. Suppose that πΎ, πΎπ‘ , π’π‘ β πΏ2 (π»2 (Ξ©)), π2 π’/ππ‘2 , π4 π’/ππ‘4 β πΏ2 (πΏ2 (Ξ©)), πΎ β πΏβ (π»2 (Ξ©)), and π3 π’/ππ‘3 β πΏβ (πΏ2 (Ξ©)); then there exists a constant πΆ > 0 independent of the spatial mesh parameter β and time discretization parameter Ξπ‘ such that σ΅©σ΅© π½ σ΅© σ΅© π½σ΅© 2βπ σ΅© σ΅© σ΅© σ΅©σ΅©σ΅©πΎ β πΎβ σ΅©σ΅©σ΅©π β€ πΆβ (σ΅©σ΅©σ΅©πΎσ΅©σ΅©σ΅©πΏβ (π»2 ) + σ΅©σ΅©σ΅©πΎσ΅©σ΅©σ΅©πΏ2 (π»2 ) π’ at π‘ = 1 0.2 0.15 σ΅© σ΅© σ΅© σ΅© +σ΅©σ΅©σ΅©π’π‘ σ΅©σ΅©σ΅©πΏ2 (π»2 ) + σ΅©σ΅©σ΅©πΎπ‘ σ΅©σ΅©σ΅©πΏ2 (π»2 ) ) 0.1 1 0.05 0 1 0.8 0.6 0.8 0.6 0.4 0.4 0.2 02 0.2 0 0 Figure 1: Surface for exact solution π’. σ΅©σ΅© 3 σ΅©σ΅© σ΅©σ΅© 2 σ΅©σ΅© σ΅©π π’σ΅© σ΅©π π’σ΅© + σ΅©σ΅©σ΅©σ΅© 3 σ΅©σ΅©σ΅©σ΅© + Ξπ‘ (σ΅©σ΅©σ΅©σ΅© 2 σ΅©σ΅©σ΅©σ΅© σ΅©σ΅© ππ‘ σ΅©σ΅©πΏ2 (πΏ2 ) σ΅©σ΅© ππ‘ σ΅©σ΅©πΏβ (πΏ2 ) σ΅©σ΅© 4 σ΅©σ΅© σ΅©π π’σ΅© +σ΅©σ΅©σ΅©σ΅© 4 σ΅©σ΅©σ΅©σ΅© ) , π = 0, 1. σ΅©σ΅© ππ‘ σ΅©σ΅©πΏ2 (πΏ2 ) Proof. Use (39a)β(39d) to get ( ππ β ππβ1 = β( , Vβ ) + (π 1π , Vβ ) , Ξπ‘ π’β at π‘ = 1, Ξπ‘ = 1/16 ( 0.15 0.1 1 0.05 ( 0.8 0.6 0.8 0.6 0.4 0.2 0.2 0 0 Figure 2: Surface for numerical solution π’β . ( βVβ β πβ , (49c) βzβ β Wβ , βππ β βππβ1 ππ β ππβ1 , wβ ) β ( , wβ ) Ξπ‘ Ξπ‘ = 0, (49b) βπβ β πβ , ππ ππ β ππβ1 ππβ1 ππ β ππβ1 , zβ ) + ( , zβ ) Ξπ‘ Ξπ‘ = 0, 0.4 (49a) ππ β ππβ1 ππ β ππβ1 , πβ ) + ( , βπβ ) Ξπ‘ Ξπ‘ ππ β ππβ1 = β( , πβ ) , Ξπ‘ 0.2 0 1 ππ β ππβ1 , Vβ ) + (ππ βππ , βVβ ) Ξπ‘ (48) βwβ β Wβ . (49d) Advances in Mathematical Physics 9 π1 (left) and π2 (right) at π‘ = 1 1 1 0.5 0.5 0 0 1 β0.5 1 β0.5 β1 1 β1 1 0.5 0.5 0.5 0.5 0 0 0 0 Figure 3: Surface for exact solution π = (π1 , π2 ). π1β (left) and π2β (right) at π‘ = 1, Ξπ‘ = 1/16 1 1 0.5 0.5 0 0 1 β0.5 β1 1 0.5 1 β0.5 β1 1 0.5 0.5 0.5 0 0 0 0 Figure 4: Surface for numerical solution πβ = (π1β , π2β ). Taking Vβ = (ππ β ππβ1 )/Ξπ‘, πβ = (ππ β ππβ1 )/Ξπ‘, zβ = (βππ β βππβ1 )/Ξπ‘, and wβ = (βππ β βππβ1 )/Ξπ‘, respectively, in (49a)β (49d), we get 1 σ΅©σ΅©σ΅© π 1/2 π σ΅©σ΅©σ΅©2 σ΅©σ΅©σ΅© πβ1 1/2 πβ1 σ΅©σ΅©σ΅©2 (σ΅©(π ) βπ σ΅©σ΅© β σ΅©σ΅©σ΅©(π ) βπ σ΅©σ΅©σ΅© ) σ΅© σ΅© 2Ξπ‘ σ΅©σ΅© σ΅© 1 σ΅©σ΅© π σ΅©2 σ΅©σ΅©π (βππ β βππβ1 )σ΅©σ΅©σ΅© + σ΅© 2Ξπ‘ σ΅© = (ππ βππ , β + ππ β ππβ1 ) Ξπ‘ 1 ππ β ππβ1 πβ1 ( βπ , βππβ1 ) 2 Ξπ‘ = β( ππ β ππβ1 ππ β ππβ1 , ) Ξπ‘ Ξπ‘ 10 Advances in Mathematical Physics π1 (left) and π2 (right) at π‘ = 1 0.5 0.5 0 0 1 1 β0.5 1 0.5 β0.5 1 0.5 0.5 0.5 0 0 0 0 Figure 5: Surface for exact solution π = (π1 , π2 ). Table 3: (πΏ2 )2 -norm and convergence order for π and π. (β, Ξπ‘) βπ β πβ βπΏβ ((πΏ2 (Ξ©))2 ) βπ β πβ βπΏβ ((πΏ2 (Ξ©))2 ) Order Order (β2/8, 1/8) 2.3958π β 001 (β2/16, 1/16) 1.3011π β 001 0.8808 1.2960π β 001 0.8641 (β2/32, 1/32) 6.8501π β 002 0.9255 6.8434π β 002 0.9213 (β2/64, 1/64) 3.5250π β 002 0.9585 3.5242π β 002 0.9574 β( ππ β ππβ1 ππ β ππβ1 , ) Ξπ‘ Ξπ‘ + (π 1π , + (π 1π , ππ β ππβ1 1 ππ β ππβ1 πβ1 )+ ( βπ , βππβ1 ) Ξπ‘ 2 Ξπ‘ ππ β ππβ1 ππ β ππβ1 ππ β ππβ1 βππ β βππβ1 =( , )+( , ) Ξπ‘ Ξπ‘ Ξπ‘ Ξπ‘ + + (π 1π , +( σ΅©σ΅© π πβ1 σ΅© σ΅©σ΅©2 σ΅© 1/2 βπ β βπ σ΅©σ΅© = βσ΅©σ΅©σ΅©σ΅©(ππ ) σ΅©σ΅© Ξπ‘ σ΅©σ΅© σ΅©σ΅© ππ β ππβ1 πβ1 βππ β βππβ1 π , ) Ξπ‘ Ξπ‘ π +( ππ β ππβ1 ππ β ππβ1 , ) Ξπ‘ Ξπ‘ πβ1 πβ1 π βπ Ξπ‘ π 1 ππ β ππβ1 πβ1 ( βπ , βππβ1 ) 2 Ξπ‘ β( βππβ1 , βππβ1 ) ππ (ππ β ππβ1 ) βππ β βππβ1 = β( , ) Ξπ‘ Ξπ‘ β( + ππ β ππβ1 ππ β ππβ1 ππ β ππβ1 )β( , ) Ξπ‘ Ξπ‘ Ξπ‘ ππ β ππβ1 ππ β ππβ1 ππ β ππβ1 )β( , ) Ξπ‘ Ξπ‘ Ξπ‘ 1 π βπ ( 2 Ξπ‘ π 2.3589π β 001 π βπ Ξπ‘ ππβ1 , πβ1 , π βπ β βπ Ξπ‘ ππ β ππβ1 ) Ξπ‘ πβ1 ) + (π 1π , β( + ππ β ππβ1 ) Ξπ‘ ππ β ππβ1 ππ β ππβ1 , ) Ξπ‘ Ξπ‘ 1 ππ β ππβ1 πβ1 ( βπ , βππβ1 ) . 2 Ξπ‘ (50) Advances in Mathematical Physics 11 π1β (left) and π2β (right) at π‘ = 1, Ξπ‘ = 1/16 0.5 0.5 0 0 1 1 β0.5 1 β0.5 1 0.5 0.5 0.5 0.5 0 0 0 0 Figure 6: Surface for numerical solution πβ = (π1β , π2β ). Multiply (51) by 2Ξπ‘, sum from π = 1 to π½, and use (19) to get By using (50), we obtain 1 σ΅©σ΅©σ΅© π 1/2 π σ΅©σ΅©σ΅©2 σ΅©σ΅©σ΅© πβ1 1/2 πβ1 σ΅©σ΅©σ΅©2 (σ΅©(π ) βπ σ΅©σ΅© β σ΅©σ΅©σ΅©(π ) βπ σ΅©σ΅©σ΅© ) σ΅© σ΅© 2Ξπ‘ σ΅©σ΅© σ΅© + π½ σ΅© σ΅©σ΅© βππ β βππβ1 σ΅©σ΅©σ΅©2 σ΅© σ΅©2 σ΅©σ΅© π0 σ΅©σ΅©σ΅©σ΅©βππ½ σ΅©σ΅©σ΅©σ΅© + 2π0 Ξπ‘ β σ΅©σ΅©σ΅©σ΅© σ΅©σ΅© Ξπ‘ σ΅© σ΅©σ΅© π=1σ΅© 1 σ΅©σ΅© π σ΅©2 σ΅©σ΅©π (βππ β βππβ1 )σ΅©σ΅©σ΅© σ΅© 2Ξπ‘ σ΅© σ΅©σ΅© σ΅©σ΅© π σ΅© π½ σ΅© π β ππβ1 πβ1 σ΅©σ΅©σ΅© σ΅©σ΅©σ΅© βππ β βππβ1 σ΅©σ΅©σ΅© σ΅©σ΅© β€ 2Ξπ‘ β (σ΅©σ΅©σ΅©σ΅© π σ΅©σ΅©σ΅© σ΅©σ΅©σ΅© σ΅©σ΅© Ξπ‘ σ΅©σ΅© σ΅©σ΅© σ΅© Ξπ‘ σ΅© π=1 σ΅© σ΅©σ΅©σ΅© ππ β ππβ1 σ΅©σ΅©σ΅© σ΅©σ΅©σ΅© ππ β ππβ1 σ΅©σ΅©σ΅© σ΅©σ΅© σ΅©σ΅© σ΅©σ΅©) + σ΅©σ΅©σ΅©σ΅© σ΅©σ΅© σ΅©σ΅© σ΅© Ξπ‘ σ΅©σ΅© σ΅©σ΅© σ΅©σ΅© Ξπ‘ σ΅©σ΅©σ΅© σ΅© π π β π πβ1 σ΅©σ΅© π½ σ΅© σ΅© σ΅©σ΅© σ΅© σ΅© 1 σ΅© σ΅©σ΅© σ΅©σ΅©σ΅©ππβ1 σ΅©σ΅©σ΅© + 2 σ΅©σ΅©σ΅©σ΅©ππ½ σ΅©σ΅©σ΅©σ΅© σ΅©σ΅©σ΅©σ΅©π 1π½ σ΅©σ΅©σ΅©σ΅© + 2Ξπ‘ β σ΅©σ΅©σ΅©σ΅© 1 σ΅©σ΅© σ΅©σ΅© σ΅©σ΅© Ξπ‘ σ΅©σ΅© σ΅© π=1 σ΅© σ΅©σ΅© π½ σ΅© σ΅© π β ππ½β1 σ΅©σ΅©σ΅© σ΅©σ΅© π½ σ΅©σ΅© + 2 σ΅©σ΅©σ΅©σ΅© πσ΅©σ΅©σ΅© σ΅©σ΅©σ΅©π σ΅©σ΅©σ΅© σ΅©σ΅© Ξπ‘ σ΅©σ΅© σ΅© π½ σ΅© σ΅©σ΅© ππ β 2ππβ1 + ππβ2 σ΅©σ΅©σ΅© σ΅©σ΅© πβ1 σ΅©σ΅© σ΅©σ΅© σ΅©σ΅©π σ΅©σ΅© + 2Ξπ‘ β σ΅©σ΅©σ΅©σ΅© σ΅©σ΅© σ΅© σ΅© (Ξπ‘)2 σ΅© σ΅© π=1 σ΅© σ΅©σ΅© σ΅©2 σ΅©σ΅© π 1/2 βππ β βππβ1 σ΅©σ΅©σ΅© σ΅© σ΅©σ΅© + σ΅©σ΅©(π ) σ΅©σ΅© Ξπ‘ σ΅©σ΅© σ΅© = β( ππ β ππβ1 πβ1 ππ β ππβ1 π ,β ) Ξπ‘ Ξπ‘ +( + (ππ , π 1π ) β (ππβ1 , π 1πβ1 ) Ξπ‘ β( β π 1π π 1πβ1 β Ξπ‘ , ππβ1 ) π½ ((ππ β ππβ1 ) /Ξπ‘, ππ ) β ((ππβ1 β ππβ2 ) /Ξπ‘, ππβ1 ) Ξπ‘ +( + ππ β ππβ1 ππ β ππβ1 , ) Ξπ‘ Ξπ‘ ππ β 2ππβ1 + ππβ2 (Ξπ‘)2 σ΅© σ΅©2 + πΆΞπ‘ β σ΅©σ΅©σ΅©σ΅©βππβ1 σ΅©σ΅©σ΅©σ΅© π=1 σ΅© σ΅©2 σ΅©σ΅© πβ1 σ΅©σ΅© σ΅©σ΅©σ΅© ππ β ππβ1 σ΅©σ΅©σ΅© σ΅© σ΅©σ΅© σ΅© σ΅© β€ πΆΞπ‘ β (σ΅©σ΅©π σ΅©σ΅© + σ΅©σ΅© σ΅©σ΅© Ξπ‘ σ΅©σ΅© σ΅© π=1 π½ , ππβ1 ) σ΅©2 σ΅© π σ΅©σ΅© π πβ1 σ΅©2 σ΅© π β 2ππβ1 + ππβ2 σ΅©σ΅©σ΅© σ΅©σ΅©σ΅© π 1 β π 1 σ΅©σ΅©σ΅© σ΅© σ΅© σ΅©σ΅© ) + σ΅©σ΅©σ΅©σ΅© + σ΅©σ΅© σ΅©σ΅© σ΅©σ΅© Ξπ‘ σ΅©σ΅© σ΅©σ΅© σ΅©σ΅© (Ξπ‘)2 σ΅© 1 ππ β ππβ1 πβ1 ( βπ , βππβ1 ) . 2 Ξπ‘ (51) π½ σ΅© σ΅©σ΅© βππ β βππβ1 σ΅©σ΅©σ΅©2 σ΅©σ΅© + π0 Ξπ‘ β σ΅©σ΅©σ΅©σ΅© σ΅©σ΅© Ξπ‘ σ΅©σ΅© σ΅© π=1σ΅© 12 Advances in Mathematical Physics σ΅© σ΅©2 π σ΅© σ΅©2 σ΅©σ΅© π½ σ΅©σ΅©2 σ΅©σ΅©σ΅© ππ½ β ππ½β1 σ΅©σ΅©σ΅© σ΅© σ΅©σ΅© ) + 0 σ΅©σ΅©σ΅©βππ½ σ΅©σ΅©σ΅© σ΅© σ΅© + πΆ (σ΅©σ΅©π 1 σ΅©σ΅© + σ΅©σ΅© σ΅© σ΅© σ΅© Ξπ‘ 2 σ΅©σ΅© σ΅©σ΅© Use (55) and PoincareΜ inequality (19) to get π‘π½ σ΅©σ΅© π½ σ΅©σ΅©2 σ΅©σ΅©π σ΅©σ΅© β€ πΆ (σ΅©σ΅©σ΅©σ΅©πσ΅©σ΅©σ΅©σ΅©2πΏβ πΏ2 + β« (σ΅©σ΅©σ΅©σ΅©ππ‘ (π )σ΅©σ΅©σ΅©σ΅©2 + σ΅©σ΅©σ΅©σ΅©ππ‘ (π )σ΅©σ΅©σ΅©σ΅©2 ) ππ ( ) σ΅© σ΅© π‘ π½ σ΅© σ΅©2 + πΆΞπ‘ β σ΅©σ΅©σ΅©σ΅©βππβ1 σ΅©σ΅©σ΅©σ΅© . 0 π=1 (52) σ΅©σ΅© 3 σ΅©σ΅©2 σ΅©σ΅© 2 σ΅©σ΅©2 σ΅©π π’σ΅© σ΅©π π’σ΅© + (Ξπ‘)2 [σ΅©σ΅©σ΅©σ΅© 3 σ΅©σ΅©σ΅©σ΅© + σ΅©σ΅©σ΅©σ΅© 2 σ΅©σ΅©σ΅©σ΅© σ΅©σ΅© ππ‘ σ΅©σ΅©πΏβ (πΏ2 ) σ΅©σ΅© ππ‘ σ΅©σ΅©πΏ2 (πΏ2 ) σ΅©σ΅© 4 σ΅©σ΅©2 σ΅©π π’σ΅© + σ΅©σ΅©σ΅©σ΅© 4 σ΅©σ΅©σ΅©σ΅© ]) . σ΅©σ΅© ππ‘ σ΅©σ΅©πΏ2 (πΏ2 ) Use Gronwall lemma to obtain σ΅© σ΅©2 π½ σ΅©σ΅© π½ σ΅©σ΅©2 σ΅© σ΅©σ΅© ππ β ππβ1 σ΅©σ΅©σ΅© σ΅© σ΅©σ΅©βπ σ΅©σ΅© β€ πΆΞπ‘ β (σ΅©σ΅©σ΅©ππβ1 σ΅©σ΅©σ΅© + σ΅©σ΅©σ΅© σ΅©σ΅© σ΅© σ΅©σ΅© σ΅© σ΅© σ΅©σ΅© σ΅© Ξπ‘ σ΅© σ΅© π=1 (56) σ΅©σ΅© π π β π πβ1 σ΅©σ΅©2 σ΅©σ΅© π πβ1 πβ2 σ΅© σ΅©2 σ΅© 1 σ΅© σ΅©σ΅© + σ΅©σ΅©σ΅© π β 2π + π σ΅©σ΅©σ΅© ) +σ΅©σ΅©σ΅©σ΅© 1 σ΅©σ΅© σ΅©σ΅© σ΅©σ΅© Ξπ‘ σ΅©σ΅© σ΅©σ΅© σ΅©σ΅© σ΅©σ΅© (Ξπ‘)2 σ΅© σ΅©2 σ΅©σ΅© π½ σ΅©σ΅©2 σ΅©σ΅©σ΅© ππ½ β ππ½β1 σ΅©σ΅©σ΅© σ΅©σ΅© ) . + πΆ (σ΅©σ΅©σ΅©π 1 σ΅©σ΅©σ΅© + σ΅©σ΅©σ΅© σ΅©σ΅© Ξπ‘ σ΅©σ΅©σ΅© (53) Note that Combining Lemmas 6-7, (55), (56) and using the triangle inequality, we complete the proof. 5. 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: π’π‘ + β β (πβ (β β (π (π‘) βπ’))) = π (x, π‘) , σ΅©σ΅© π π β σ΅©σ΅© 1 σ΅©σ΅© σ΅©σ΅© Ξπ‘ σ΅© σ΅©σ΅© π σ΅©σ΅© π’ β 2π’πβ1 + π’πβ2 π’π‘ (π‘π ) β π’π‘ (π‘πβ1 ) σ΅©σ΅©σ΅© σ΅©σ΅© β σ΅©σ΅© = σ΅©σ΅©σ΅© σ΅©σ΅© Ξπ‘ σ΅©σ΅© σ΅©σ΅© (Ξπ‘)2 σ΅© 2 π 1πβ1 σ΅©σ΅©σ΅©σ΅© σ΅© σ΅©2 σ΅©σ΅© π σ΅©σ΅©2 σ΅© π’ β 2π’πβ1 + π’πβ2 σ΅©σ΅© σ΅© = σ΅©σ΅©σ΅©σ΅© β π’ (π‘ ) π‘π‘ Ξ 2 β σ΅© σ΅©σ΅© σ΅©σ΅© (Ξπ‘) σ΅© 2 σ΅©σ΅© 3 σ΅©σ΅© σ΅©σ΅© 4 σ΅©σ΅© σ΅©π π’σ΅© σ΅©σ΅© π π’ σ΅©σ΅© σ΅©σ΅© 4 σ΅©σ΅© ππ + Ξπ‘σ΅©σ΅©σ΅© 3 σ΅©σ΅©σ΅© ] σ΅©σ΅© ππ‘ σ΅©σ΅© β 2 σ΅© ππ‘ σ΅©σ΅© π‘πβ2 σ΅© σ΅© σ΅© σ΅© σ΅©πΏ (πΏ ) π‘π σ΅©σ΅© 4 σ΅© σ΅©σ΅© 3 σ΅© σ΅©σ΅© π π’ σ΅©σ΅© σ΅©π π’σ΅© σ΅©σ΅© 4 σ΅©σ΅© ππ + (Ξπ‘)2 σ΅©σ΅©σ΅© 3 σ΅©σ΅©σ΅© σ΅© σ΅© σ΅©σ΅© ππ‘ σ΅©σ΅© β 2 , ππ‘ σ΅© σ΅© π‘πβ2 σ΅© σ΅© σ΅© σ΅©πΏ (πΏ ) β€ πΆ(Ξπ‘)2 β« π‘π π’ (x, 0) = sin (ππ₯1 ) sin (ππ₯2 ) , (x, π‘) β πΞ© × π½, x = (π₯1 , π₯2 ) β Ξ©, (57) β€ πΆ[Ξπ‘ β« σ΅©2 π’ (x, π‘) = Ξπ’ (x, π‘) = 0, (x, π‘) β Ξ© × π½, σ΅©2 where π‘Ξ β β (π‘πβ1 , π‘π ) . where Ξ© = [0, 1] × [0, 1], π½ = (0, 1], π(π‘) = 1 + π‘2 , π = 1, and π(x, π‘) is chosen so that the exact solution for the scalar unknown function is π’ (x, π‘) = πβ2π‘ sin (ππ₯1 ) sin (ππ₯2 ) , (58) its exact gradient is π = βπ’ = πβ2π‘ π [ (54) cos (ππ₯1 ) sin (ππ₯2 ) sin (ππ₯1 ) cos (ππ₯2 ) ], (59) its exact flux function is cos (ππ₯1 ) sin (ππ₯2 ) ], π = βππ = πβ2π‘ (1 + π‘2 ) π [ sin (ππ₯1 ) cos (ππ₯2 ) Substitute (44), (47) and (54) into (53) to obtain π‘π½ σ΅©σ΅© π½ σ΅©σ΅©2 σ΅©σ΅©βπ σ΅©σ΅© β€ πΆ (σ΅©σ΅©σ΅©σ΅©πσ΅©σ΅©σ΅©σ΅©2πΏβ πΏ2 + β« (σ΅©σ΅©σ΅©σ΅©ππ‘ (π )σ΅©σ΅©σ΅©σ΅©2 + σ΅©σ΅©σ΅©σ΅©ππ‘ (π )σ΅©σ΅©σ΅©σ΅©2 ) ππ ( ) σ΅© σ΅© π‘ 0 σ΅©σ΅© 2 σ΅©σ΅©2 σ΅©σ΅© 3 σ΅©σ΅©2 σ΅©π π’σ΅© σ΅©π π’σ΅© + σ΅©σ΅©σ΅©σ΅© 2 σ΅©σ΅©σ΅©σ΅© + (Ξπ‘) [σ΅©σ΅©σ΅©σ΅© 3 σ΅©σ΅©σ΅©σ΅© σ΅©σ΅© ππ‘ σ΅©σ΅©πΏβ (πΏ2 ) σ΅©σ΅© ππ‘ σ΅©σ΅©πΏ2 (πΏ2 ) (60) and the diffusion term is πΎ = β β π = 2π2 πβ2π‘ (1 + π‘2 ) sin (ππ₯1 ) sin (ππ₯2 ) . 2 σ΅©σ΅© 4 σ΅©σ΅©2 σ΅©π π’σ΅© +σ΅©σ΅©σ΅©σ΅© 4 σ΅©σ΅©σ΅©σ΅© ]) . σ΅©σ΅© ππ‘ σ΅©σ΅©πΏ2 (πΏ2 ) (55) (61) Dividing the domain Ξ© into the triangulations of mesh size β uniformly and choosing the piecewise linear space πβ with index π = 1 and β = β2Ξπ‘ = β2/8, β2/16, β2/32, β2/64, we obtain the optimal a priori error estimates in πΏ2 and π»1 -norm for the scalar unknown π’ in Table 1. Similarly, from Table 2, we can find that both βπΎ β πΎβ βπΏβ (π»1 (Ξ©)) and Advances in Mathematical Physics 13 (πΏ2 (Ξ©))2 space; therefore, the regularity requirement on the gradient solution π = βπ’ is reduced. We obtain the optimal priori error estimates in πΏ2 and π»1 -norm for both the scalar unknown π’ and the diffusion term πΎ and the priori error estimates in (πΏ2 )2 -norm for its gradient π and its flux π. 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 [29β 32, 38] for solving fourth-order nonlinear elliptic equation and fourth-order nonlinear parabolic equations. πΎ at π‘ = 1 6 4 1 2 0 1 0.8 0.6 0.8 0.4 0.6 0.4 0.2 0.2 0 0 Figure 7: Surface for exact solution πΎ. Acknowledgments πΎβ at π‘ = 1, Ξπ‘ = 1/16 6 4 1 2 The authors thank the referees and editor for their valuable suggestions and comments, which greatly improve the paper. This work is supported by National Natural Science Fund (11061021), Natural Science Fund of Inner Mongolia Autonomous Region (2012MS0108, 2012MS0106), Scientific Research Projection of Higher Schools of Inner Mongolia (NJZZ12011, NJ10006, NJZY13199), and Program of Higherlevel talents of Inner Mongolia University (125119, 30105125132). 0.8 0 1 0.6 0.8 0.6 0.4 0.4 0.2 0.2 0 0 Figure 8: Surface for numerical solution πΎβ . βπΎ β πΎβ βπΏβ (πΏ2 (Ξ©)) for the diffusion term πΎ are optimal, too. At the same time, we take the corresponding piecewise constant space Wβ with index π = 0 for the gradient π and the flux π and obtain some convergence results in (πΏ2 )2 -norm with β = β2Ξπ‘ = β2/8, β2/16, β2/32, β2/64 in Table 3. Taking π‘ = 1 with β = β2Ξπ‘ = β2/32, we show the surfaces of the exact solution π’, π, π, and πΎ in Figures 1, 3, 5, and 7, respectively. 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