# A New Linearized Crank-Nicolson Mixed Element Scheme for the Extended Fisher-Kolmogorov Equation

Authors: Jinfeng Wang; Hong Li; Siriguleng He; Wei Gao; Yang Liu
Journal: The Scientific World Journal
 (2013)

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## Abstract

We present a new mixed finite element method for solving the extended Fisher-Kolmogorov (EFK) equation. We first decompose the EFK equation as the two second-order equations, then deal with a second-order equation employing finite element method, and handle the other second-order equation using a new mixed finite element method. In the new mixed finite element method, the gradient∇u belongs to the weaker (L2(Ω))2 space taking the place of the classical H(div;Ω) space. We prove some a priori bounds for the solution for semidiscrete scheme and derive a fully discrete mixed scheme based on a linearized Crank-Nicolson method. At the same time, we get the optimal a priori error estimates in L2 and H1-norm for both the scalar unknown u and the diffusion term w=−Δu and a priori error estimates in (L2)2-norm for its gradient χ=∇u for both semi-discrete and fully discrete schemes.


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## Body

## 1. Introduction

In recent years, a lot of numerical methods for solving fourth-order partial differential equations have been presented and analyzed by many researchers. In [1–6], authors studied some mixed finite element methods for fourth-order elliptic and parabolic equations. Shi et al. [7], Wang et al. [8, 9], and H. R. Chen and S. C. Chen [10] proposed some nonconforming finite element methods for fourth-order elliptic equation (or biharmonic equation). In [11], some rectangular finite element methods for fourth-order elliptic singular perturbation problems were considered. Hu and Shi [12] studied the best L2 norm error estimate of lower order finite element methods for the fourth-order problem. Chen and Wang [13] discussed a mixed finite element method for thin film epitaxy. In [14], a conforming finite element approximation for the fourth-order Steklov eigenvalue problem is discussed. In [15], a Crank-Nicolson time-stepping was used to approximate the differential term and the product trapezoidal method was employed to treat the integral term, and the quasi-wavelets numerical method for space discretization. Some numerical methods were proposed and studied for Cahn-Hilliard equations, such as (mixed) finite element methods [16–18], time-stepping methods [19, 20], spectral Galerkin method [21], discontinuous Galerkin method [22], and a conservative nonlinear difference scheme [23]. Liu et al. [24] studied a C1-conforming finite element method for a fourth-order nonlinear hyperbolic equation. He et al. [25], proposed some mixed element schemes for fourth-order linear wave equation. In [26–28], some (mixed) finite element methods were studied for the extended Fisher-Kolmogorov equations.In [29, 30], authors proposed and discussed a new mixed finite element scheme based on new mixed spaces for second-order linear elliptic equations. In the new mixed method, a weaker (L2(Ω))2 space was provided to replace the complex H(div;Ω) space. Considering the advantage of the new mixed method, some problems have been studied, such as second-order Sobolev equation [31], second-order parabolic equations [32, 33], and reaction-convection-diffusion problems [34, 35]. However, the new mixed element method for the extended Fisher-Kolmogorov equation has not been studied in the literatures.In this paper, our aim is to apply the new mixed method [29, 30] to solve the extended Fisher-Kolmogorov (EFK) equation [28]
(1)ut+γΔ2u-Δu+f(u)=0,(x,t)∈Ω×J,with initial condition(2)u(x,0)=u0(x),x∈Ω,and Dirichlet boundary conditions(3)u(x,t)=Δu(x,t)=0,(x,t)∈∂Ω×J-,whereΩ is a bounded convex polygonal domain in Rd, d≤2 with boundary ∂Ω, and  J=(0,T] is the time interval with 0<T<∞. u0(x) is given function, f(u)=u3-u coefficient γ is a positive constant.For formulating the new mixed scheme, we first introduce a diffusion termw=-Δu to get the following two lower order equations: (4a)ut-γΔw+w=-f(u),(x,t)∈Ω×J,(4b)-Δu=w,(x,t)∈Ω×J.Then, we introduce another auxiliary variableχ=∇u in (4b) to obtain the following lower-order system: (5a)ut-γΔw+w=-f(u),(5b)-∇·χ=w,(5c)χ-∇u=0.For the lower-order system (5a), (5b), and (5c) we will formulate a new mixed scheme based on [29, 30] and discuss some a priori bounds for the mixed element solution and a priori error estimates for semidiscrete scheme. What is more, we will get the fully discrete scheme based on linearized Crank-Nicolson method and some a priori error estimates. From the obtained results, we can find easily that a priori error estimates in L2 and H1-norm for the scalar unknown u and the diffusion term w are optimal. Moreover, we also prove a priori error estimate in (L2)2-norm for the gradient term χ.The remainder of this paper is organized as follows. In Section2, a new mixed weak formulation and semidiscrete mixed scheme are formulated for system (5a), (5b), and (5c), then, some a priori bounds for the new mixed element solution are derived. In Section 3, a new mixed elliptic projection operator 𝒬h and a projection operator 𝒮h associated with the coupled systems are presented and some a priori error estimates for semidiscrete scheme are proved. In Section 4, some a priori error estimates for fully discrete linearized Crank-Nicolson scheme are analyzed. Finally in Section 5, some concluding remarks and extensions about the new mixed finite element method are given. In this paper, C>0 is a generic constant which does not depend on the spatial mesh parameter h or time step parameter Δt. At the same time, we denote the natural inner product in L2(Ω) or (L2(Ω))2 by  (·,·) with the corresponding norm ∥·∥. The others notations and definitions of Sobolev spaces as in [36, 37] are used. In order to facilitate the expression, we denote B=B(Ω), F=(F(Ω))d, such as H01≜H01(Ω), L2≜(L2(Ω))2, H1≜(H1(Ω))2.


## 2. New Mixed Weak Formulation and Semidiscrete Scheme

In the following analysis, we will discuss the new mixed formulation and some a priori error estimates for semidiscrete scheme. Based on the lower-order system (5a), (5b), and (5c) the new mixed weak formulation is to find {u,w,χ}:[0,T]↦H01×H01×L2 such that (6a)(ut,v)+γ(∇w,∇v)+(w,v)+(f(u),v)=0,∀v∈H01,(6b)(χ,∇g)=(w,g),∀g∈H01,(6c)(χ,z)-(∇u,z)=0,∀z∈L2.LetHh and Lh, respectively, be finite dimensional subspaces of H01 and L2 defined by the finite element pair P1-P02 [29, 30]
(7)Hh={vh∈C0(Ω)∩H01∣vh∈P1(K),∀K∈𝒦h},Lh={wh=(w1h,w2h)∈L2∣w1h,w2h∈P0(K),∀K∈𝒦h}.Then, the semidiscrete mixed scheme for (6a), (6b), and (6c) is to determine {uh,wh,χh}:[0,T]↦Hh×Hh×Lh such that (8a)(uht,vh)+γ(∇wh,∇vh)+(wh,vh)+(f(uh),vh)=0,∀vh∈Hh,(8b)(χh,∇gh)=(wh,gh),∀gh∈Hh,(8c)(χh,zh)-(∇uh,zh)=0,∀zh∈Lh.In the following discussion, we derive some a priori bounds for the finite element solution.Theorem 1.
Let{uh,wh,χh} be the solution for system (8a), (8b), and (8c) then there exists a constant C>0 such that, for 1≤p<+∞(9)∥uh∥Lp+∥uh∥1+∥wh∥+∥χh∥+(∫0t‍∥uht∥2ds)1/2≤C(γ,∥u0h∥,∥w0h∥).Proof.
Differentiating (8b) and (8c) with respect to t, we obtain (10a)(uht,vh)+γ(∇wh,∇vh)+(wh,vh)+(f(uh),vh)=0,∀vh∈Hh,(10b)-(χht,∇gh)+(wht,gh),∀gh∈Hh,(10c)(χht,zh)-(∇uht,zh)=0,∀zh∈Lh.
Takevh=uht, gh=wh, zh=∇wh in (10a), (10b), and (10c) to obtain (11a)∥uht∥2+γ(∇wh,∇uht)+(wh,uht)+(f(uh),uht)=0,(11b)-γ(χht,∇wh)+γ2ddt∥wh∥2=0,(11c)γ(χht,∇wh)-γ(∇uht,∇wh)=0.
We defineF(σ)≜(1/4)(1-σ2)2 and add the three equations for system (11a), (11b), and (11c) to get
(12)ddt(γ2∥wh∥2+(F(uh),1))+∥uht∥2=-(wh,uht)≤12∥wh∥2+12∥uht∥2.
Integrate (12) with respect to time from 0 to t and use Gronwall lemma to have
(13)γ2∥wh∥2+(F(uh),1)+∫0t‍∥uht∥2ds≤C(γ∥wh(0)∥2+(F(uh(0)),1)).
We note thatF(uh)≥0 to obtain
(14)∥uh∥+∥wh∥≤C(γ,∥u0h∥,∥w0h∥).
Takegh=uh in (8b) and zh=∇uh in (8c) to obtain
(15)∥∇uh∥≤C(∥uh∥+∥wh∥).
Substitute (14) into (15) to obtain
(16)∥∇uh∥≤C(γ,∥u0h∥,∥w0h∥).
Choosezh=χh in (8c) and use (16) to obtain
(17)∥χh∥≤∥∇uh∥≤C(γ,∥u0h∥,∥w0h∥).
ForΩ⊂R2, we use Sobolev imbedding theorem to get
(18)∥uh∥Lp≤∥uh∥1≤C(γ,∥u0h∥,∥w0h∥).
By (14), (16)–(18), we accomplish the proof of Theorem 1.Remark 2.
Based on the discussion in [29, 30], we can see easily that the discrete LBB condition for the mixed finite element space (Hh,Lh) is satisfied. In our weak formulation (6a), (6b), and (6c) the H(div;Ω) space is replaced by the weaker L2 space. Compared to the H(div;Ω) space, the regularity requirement for our L2 space is reduced.


## 3. A Priori Error Estimates for Semidiscrete Scheme

In order to analyze the convergence of the method, we first introduce the projection operator𝒮h and a new mixed elliptic projection operator 𝒬h associated with the coupled equations.Lemma 3.
Let𝒮h:[0,T]→H01→Hh be satisfied by the following relation:
(19)(∇(w-𝒮hw),∇vh)=0,∀vh∈Hh,∥w-𝒮hw∥L2(Ω)+h∥w-𝒮hw∥H1≤Ch2∥w∥H2.Lemma 4.
Let(𝒬hu,𝒬hχ):[0,T]→Hh×Lh be given by the following new mixed relations: (20a)-(χ-𝒬hχ,∇gh)=0,∀gh∈Hh,(20b)(χ-𝒬hχ,zh)-(∇(u-𝒬hu),zh)=0,∀zh∈Lh;
there exists a constantC>0 independent of h such that
(21)∥u-𝒬hu∥+h(∥χ-𝒬hχ∥+∥u-𝒬hu∥1)≤Ch2(∥u∥H2+∥χ∥H1),∥ut-𝒬hut∥≤Ch2(∥ut∥H2+∥χt∥H1),from [29, 30], we can obtain the proof for Lemma 4.Further, assuming that the triangulation is quasi-uniform, the following error estimates [27] hold, for j=0,1 and p=2,∞:
(22)∥𝒬hu∥Wj,p≤∥τ∥Wj,p+∥u∥Wj,p≤Cu.Based on Lemmas3 and 4, the following theorem for semidiscrete error estimates is obtained.Theorem 5.
Withuh(0)=𝒬hu0, assume that the solution's regular properties for system (6a), (6b), and (6c) satisfy w,u,ut∈L∞(H2), χ,χt∈L∞(H1). Then there exists a positive constant C>0 independent of h such that
(23)∥u-uh∥+∥w-wh∥+h(∥u-uh∥1+∥χ-χh∥)≤Ch2[(∫0t‍)2∥u∥L∞(H2)+∥χ∥L∞(H1)+(∫0t‍(∥u∥H22+∥ut∥H22+∥χ∥H12+∥χt∥H12+∥w∥H22)ds∫0t‍)1/2].Proof.
To get a priori error estimates, we decompose the errors as(24)w-wh=w-𝒮hw+𝒮hw-wh=π+κ;u-uh=u-𝒬hu+𝒬hu-uh=τ+ϑ;χ-χh=χ-𝒬hχ+𝒬hχ-χh=λ+ζ.
Combining (6a)–(8c), (19) and (20a) and (20b), we can get the error equations (25a)(ϑt,vh)+γ(∇κ,∇vh)+(κ,vh)+(f(u)-f(uh),vh)=-(τt,vh)-(π,vh),∀vh∈Hh,(25b)-(ζ,∇gh)+(κ,gh)=-(π,gh),∀gh∈Hh,(25c)(ζ,zh)-(∇ϑ,zh)=0,∀zh∈Lh.
Differentiating (25b) and (25c) with respect to t, we get (26a)(ϑt,vh)+γ(∇κ,∇vh)+(κ,vh)+(f(u)-f(uh),vh)=-(τt,vh)-(π,vh),∀vh∈Hh,(26b)-(ζt,∇gh)+(κt,gh)=-(πt,gh),∀gh∈Hh,(26c)(ζt,zh)-(∇ϑt,zh)=0,∀zh∈Lh.
Choosevh=ϑt, gh=κ, and zh=∇κ in (26a), (26b), and (26c), respectively, to obtain (27a)∥ϑt∥2+γ(∇κ,∇ϑt)+(κ,ϑt)+(f(u)-f(uh),ϑt)=-(τt,ϑt)-(π,ϑt),(27b)12ddt∥κ∥2-(ζt,∇κ)=-(πt,κ),(27c)(ζt,∇κ)-(∇ϑt,∇κ)=0.
Using (27a), (27b), and (27c) and Cauchy-Schwarz inequality and Young inequality, we have
(28)∥ϑt∥2+(f(u)-f(uh),ϑt)=-γ(∇κ,∇ϑt)-(κ,ϑt)-(τt,ϑt)-(π,ϑt)=-γ(ζt,∇κ)-(κ,ϑt)-(τt,ϑt)-(π,ϑt)=-γ2ddt∥κ∥2-(πt,κ)-(κ,ϑt)-(τt,ϑt)-(π,ϑt)≤-γ2ddt∥κ∥2+C(∥πt∥2+∥τt∥2+∥π∥2)+∥κ∥2+14∥ϑt∥2.
Using Cauchy-Schwarz inequality, Young inequality, Sobolev imbedding theorem, and inequality (22), we use the similar method as the one in [27] to obtain
(29)-(f(u)-f(uh),ϑt)=-(f(u)-f(𝒬hu),ϑt)-(f(𝒬hu)-f(uh),ϑt)=-(f(u)-f(𝒬hu),ϑt)-(f(ϑ),ϑt)-3(uh𝒬huϑ,ϑt)≤|(τ(u2+u𝒬hu+(𝒬hu)2-1),ϑt)|-ddt(F(ϑ),1)-3(uh𝒬huϑ,ϑt)≤C(∥τ∥2+∥ϑ∥2)-ddt(F(ϑ),1)+14∥ϑt∥2.
Substituting (29) into (28), we have
(30)ddt[γ2∥κ∥2+(F(ϑ),1)]+12∥ϑt∥2≤C(∥τt∥2+∥τ∥2+∥π∥2+∥πt∥2+∥ϑ∥2+∥κ∥2).
Integrate (30) with respect to time from 0 to t to obtain
(31)γ2∥κ∥2+(F(ϑ),1)+12∫0t‍∥ϑt∥2ds≤C∫0t‍(∥τt∥2+∥τ∥2+∥π∥2+∥πt∥2+∥ϑ∥2+∥κ∥2)ds.
Using Gronwall lemma, we have(32)γ2∥κ∥2+∥ϑ∥2+12∫0t‍∥ϑt∥2ds≤C∫0t‍(∥τt∥2+∥τ∥2+∥π∥2+∥πt∥2)ds.
Takinggh=ϑ in (25b) and zh=∇ϑ in (25c), we have (33a)(ζ,∇ϑ)-(κ,ϑ)=(π,ϑ),(33b)∥∇ϑ∥2-(ζ,∇ϑ)=0.
Add the two equations to get(34)∥∇ϑ∥2≤C(∥κ∥2+∥ϑ∥2+∥π∥2).
Choosezh=ζ in (25c) to obtain
(35)∥ζ∥2≤∥∇ϑ∥2≤C(∥κ∥2+∥ϑ∥2+∥π∥2).
Substitute (32) into (34) and (35) to get
(36)∥ζ∥2+∥∇ϑ∥2≤C(∥π∥2+∫0t‍(∥τt∥2+∥τ∥2+∥π∥2+∥πt∥2)ds).
Combining Lemmas3 and 4, (32)–(36) with the triangle inequality, we obtain the error estimates for Theorem 5.Remark 6.
The conclusion for Theorem5 demonstrates that the optimal convergence order in L2-norm for both the scalar unknown u and the diffusion term w is obtained. At the same time, the optimal convergence order in H1-norm for the scalar unknown u is gotten, too.Theorem 7.
Assume that the solution's regular properties for system (6a), (6b), and (6c) satisfy u,w,ut∈L∞(H2) and χ,χt∈L∞(H1). Then there exists a positive constant C independent of h such that
(37)∥w-wh∥1≤Ch[(∫0t‍)2∥u∥L∞(H2)+∥w∥L∞(H2)+∥χ∥L∞(H1)+(∫0t‍(∥u∥H22+∥ut∥H22+∥χ∥H12+∥χt∥H12+∥w∥H22)ds∫0t‍)1/2].Proof.
Choosevh=κt, gh=ϑt, and zh=ζt in (26a), (26b), and (26c), respectively, to obtain(38a)(ϑt,κt)+γ2ddt∥κ∥12+(f(u)-f(uh),κt)=-(τt,κt)-(π,κt),(38b)(ζt,∇ϑt)-(κt,ϑt)=(πt,ϑt),(38c)∥ζt∥2-(∇ϑt,ζt)=0.
Adding the three equations of system (38a), (38b), and (38c), we have
(39)γ2ddt∥κ∥12+∥ζt∥2=-(f(u)-f(uh),κt)-(τt,κt)-(π,κt)+(πt,ϑt)=-ddt(f(u)-f(uh),κ)+(f′(u)ut-f′(uh)uht,κ)-ddt(τt,κ)+(τtt,κ)-ddt(π,κ)+(πt,κ)+(πt,ϑt).
Integrate (39) with respect to t and use Cauchy-Schwarz inequality and Young inequality to get
(40)γ2∥κ∥12+∫0t‍∥ζt∥2ds=-(f(u)-f(uh),κ)+∫0t‍(f′(u)ut-f′(uh)uht,κ)ds-(τt,κ)+∫0t‍(τtt,κ)ds-(π,κ)+∫0t‍(πt,κ)ds+∫0t‍(πt,ϑt)ds≤-(f(u)-f(uh),κ)+∫0t‍(f′(u)ut-f′(uh)uht,κ)ds,C[∥τt∥2+∥π∥2+∫0t‍(∥τtt∥2+∥πt∥2+∥κ∥2+∥ϑt∥2)ds]+γ8∥κ∥2.
Using the similar method to the estimate for (29), we obtain
(41)-(f(u)-f(uh),κ)=-(f(u)-f(Rhu),κ)-(f(Rhu)-f(uh),κ)≤|(τ(u2+uRhu+(Rhu)2-1),κ)|+(f′(u^^h)(Rhu-uh),κ)=|(τ(u2+uRhu+(Rhu)2-1),κ)|+((3(u^^h)2-1)ϑ,κ)≤C(∥τ∥2+∥ϑ∥2)+γ8∥κ∥12,∫0t‍(f′(u)ut-f′(uh)uht,κ)ds=∫0t‍(f′(u)(ut-uht)+(f′(u)-f′(uh))uht,κ)ds=∫0t‍(f′(u)(ϑt+τt)+f′′(u1)(ϑ+τ)uht,κ)ds≤C∫0t‍(∥τ∥2+∥ϑ∥2+∥τt∥2+∥ϑt∥2+∥κ∥12)ds.
Substitute (41) into (40) to get
(42)γ4∥κ∥12+∫0t‍∥ζt∥2ds≤C[∥τt∥2+∥π∥2+∫0t‍(∥τ∥2+∥τt∥2+∥τtt∥2+∥πt∥2+∥κ∥12+∥ϑ∥2+∥ϑt∥2)ds].
Substitute (32) into (42) and use Gronwall lemma to get
(43)γ4∥κ∥12+∫0t‍∥ζt∥2ds≤C[∫0t‍∥τt∥2+∥π∥2+∫0t‍(∥τ∥2+∥τt∥2+∥τtt∥2+∥π∥2+∥πt∥2)ds].
Combining Lemmas3 and 4, (43) with the triangle inequality, we obtain the error estimates for Theorem 7.Remark 8.
In Theorem7, the optimal convergence order in H1-norm for the diffusion term w is obtained.


## 4. A Priori Error Estimates Based on Linearized C-N Scheme

### 4.1. Linearized Crank-Nicolson Mixed Scheme

In the following discussion, we will derive the fully discrete a priori error estimates based on a linearized Crank-Nicolson method. Let0=t0<t1<t2<⋯<tM=T be a given partition of the time interval [0,T] with step length Δt=T/M and nodes tn=nΔt, for some positive integer M. For a smooth function ϕ on [0,T], define ψn=ϕ(tn) and ψn-1/2=(ψn+ψn-1)/2.Equations (6a), (6b), and (6c) can be written as the following formulation at time t=tn-1/2: (44a)(ut(tn-1/2),v)+γ(∇w(tn-1/2),∇v)+(w(tn-1/2),v)+(f(u(tn-1/2)),v)=0,∀v∈H01,(44b)(χ(tn-1/2),∇g)-(w(tn-1/2),g)=0,∀g∈H01m,(44c)(χ(tn-1/2),z)-(∇u(tn-1/2),z)=0,∀z∈L2.Then, the following equivalent formulation for (44a), (44b), and (44c) is as follows: (45a)(un-un-1Δt,v)+γ(∇wn-1/2,∇v)+(wn-1/2,v)+(f(u^^n-1/2),v)=(R1n-1/2,v)+(R2n-1/2,v),∀v∈H01,(45b)(χn-1/2,∇g)-(wn-1/2,g)=0,∀g∈H01,(45c)(χn-1/2,z)-(∇un-1/2,z)=0,∀z∈L2,where(46)u^^n-1/2=32un-1-12un-2,R1n-1/2=un-un-1Δt-ut(tn-1/2),R2n-1/2=f(u^^n-1/2)-f(u(tn-1/2)).Now a linearized C-N fully discrete procedure is to find(uhn,whn,χhn)∈Hh×Hh×Lh,(n=0,1,…,M) such that (47a)(uhn-uhn-1Δt,vh)+γ(∇whn-1/2,∇vh)+(whn-1/2,vh)+(f(u^^hn-1/2),vh)=0,∀vh∈Hh,(47b)(χhn-1/2,∇gh)-(whn-1/2,gh)=0,∀gh∈Hh,(47c)(χhn-1/2,zh)-(∇uhn-1/2,zh)=0,∀zh∈Lh.Remark 9.
We can find that the system (47a), (47b), and (47c) is a linear scheme by a linearized term
(48)f(u^^hn-1/2)=(32uhn-1-12uhn-2)3-(32uhn-1-12uhn-2).In the following subsection, we derive a priori error estimates based on a linearized Crank-Nicolson fully discrete scheme.


### 4.2. A Priori Error Estimates for Fully Discrete Scheme

In order to derive the linearized C-N fully discrete error estimates, we now write the errors as follows:(49)w(tn)-whn=w(tn)-𝒮hwn+𝒮hwn-whn=πn+κn;u(tn)-uhn=u(tn)-𝒬hun+𝒬hun-uhn=τn+ϑn;χ(tn)-χhn=χ(tn)-𝒬hχn+𝒬hχn-χhn=λn+ζn.Subtracting (47a), (47b), and (47c) from (45a), (45b), and (45c) and using (19) and (20a) and (20b) at t=tn-1/2, we have the following error equations (50a)(ϑn-ϑn-1Δt,vh)+γ(∇κn-1/2,∇vh)+(κn-1/2,vh)+(f(u^^n-1/2)-f(u^^hn-1/2),vh)=-(τn-τn-1Δt,vh)-(πn-1/2,vh)+(R1n-1/2,vh)+(R2n-1/2,vh),∀vh∈Hh,(50b)(ζn-1/2,∇gh)-(κn-1/2,gh)=(πn-1/2,gh),∀gh∈Hh,(50c)(ζn-1/2,zh)-(∇ϑn-1/2,zh)=0,∀zh∈Lh.In order to get the fully discrete error estimates, we introduce the following lemma.Lemma 10.
ForR1n-1/2 and R2n-1/2, the following estimates hold:
(51)∥R1n-1/2∥≤CΔt2∥uttt∥L∞(L2),∥R2n-1/2∥≤C(u)Δt2∥uttt∥L∞(L2).Proof.
Using the Taylor expansion, we can obtain easily the conclusion for Lemma10.Remark 11.
From the estimate for∥R2n-1/2∥ in Lemma 10, we know that the convergence rate in time direction for the system (47a), (47b), and (47c) is order 2, so the system (47a), (47b), and (47c) is called a linearized Crank-Nicolson scheme based on the linearized term f(u^^hn-1/2).In the following discussion, we will derive some fully discrete a priori error estimates.Theorem 12.
Withuh0=𝒬hu0, assume that the solution's regular properties for system (6a), (6b), and (6c) satisfy ut∈L2(H2), uttt∈L∞(L2), u,w∈L∞(H2), χ∈L∞(H1), χt∈L2(H1). Then there exists a positive constant C independent of h and Δt such that
(52)∥u(tJ)-uhJ∥≤C[h2∥|·|∥+Δt2∥uttt∥L∞(L2)],∥|·|∥≜∥u∥L∞(H2)+∥ut∥L2(H2)+∥w∥L∞(H2)+∥χ∥L∞(H1)+∥χt∥L2(H1).Proof.
Settingvh=ϑn-1/2, gh=κn-1/2, and zh=∇κn-1/2 in (50a), (50b), and (50c), respectively, we get (53a)∥ϑn∥2-∥ϑn-1∥22Δt+γ(∇κn-1/2,∇ϑn-1/2)+(κn-1/2,ϑn-1/2)+(f(u^^n-1/2)-f(u^^hn-1/2),ϑn-1/2)=-(τn-τn-1Δt,ϑn-1/2)-(πn-1/2,ϑn-1/2)+(R1n-1/2,ϑn-1/2)+(R2n-1/2,ϑn-1/2),(53b)γ∥κn-1/2∥2-γ(ζn-1/2,∇κn-1/2)=-γ(πn-1/2,κn-1/2),(53c)γ(ζn-1/2,∇κn-1/2)-γ(∇ϑn-1/2,∇κn-1/2)=0.
Adding the three equations of (53a), (53b), and (53c) and using Cauchy-Schwarz inequality and Young inequality, we have
(54)γ∥κn-1/2∥2+12Δt(∥ϑn∥2-∥ϑn-1∥2)=-(κn-1/2,ϑn-1/2)-(f(u^^n-1/2)-f(u^^hn-1/2),ϑn-1/2)-(τn-τn-1Δt,ϑn-1/2)-(πn-1/2,ϑn-1/2)+(R1n-1/2,ϑn-1/2)+(R2n-1/2,ϑn-1/2)-γ(πn-1/2,κn-1/2)≤(∥τn-τn-1Δt∥+∥R1n-1/2∥+∥R2n-1/2∥)∥ϑn-1/2∥+∥κn-1/2∥∥ϑn-1/2∥+γ∥πn-1/2∥∥κn-1/2∥+(∥f(u^^n-1/2)-f(u^^hn-1/2)∥+∥πn-1/2∥)∥ϑn-1/2∥≤C(∥τn-τn-1Δt∥2+∥R1n-1/2∥2+∥R2n-1/2∥2+∥πn-1/2∥2)+γ2∥κn-1/2∥2+C∥f(u^^n-1/2)-f(u^^hn-1/2)∥2+C∥ϑn-1/2∥2.
In order to accomplish our process, we have to estimate the nonlinear term∥f(u^^n-1/2)-f(u^^hn-1/2)∥2. Using Young inequality, we have
(55)∥f(u^^n-1/2)-f(u^^hn-1/2)∥2≤2∥(u^^n-1/2)3-(u^^hn-1/2)3∥2+2∥u^^n-1/2-u^^hn-1/2∥2=2∥[(12)232(un-1-uhn-1)-12(un-2-uhn-2)]×[(12)2(32un-1-12un-2)2+(32uhn-1-12uhn-2)2+(32un-1-12un-2)(32uhn-1-12uhn-2)(12)2]∥2+2∥32(un-1-uhn-1)-12(un-2-uhn-2)∥2≤2∥32(ϑn-1+τn-1)-12(ϑn-2+τn-2)∥2×[(12)2∥(32un-1-12un-2)2+(32uhn-1-12uhn-2)2+(32un-1-12un-2)(32uhn-1-12uhn-2)(12)2∥L∞2+1(12)2]=(24Cu2+2)∥32(ϑn-1+τn-1)-12(ϑn-2+τn-2)∥2≤C(∥τn-1∥2+∥τn-2∥2+∥ϑn-1∥2+∥ϑn-2∥2).
Note that(56)∥τn-τn-1Δt∥2≤1Δt∫tn-1tn‍∥τt(s)∥2ds.
Substitute (55) and (56) into (54) to obtain
(57)γ2∥κn-1/2∥2+12Δt(∥ϑn∥2-∥ϑn-1∥2)≤C(1Δt∫tn-1tn‍∥τt(s)∥2ds+∥R1n-1/2∥2+∥R2n-1/2∥2+∥πn-1/2∥2∫tn-1tn‍)+C(∥τn-1∥2+∥τn-2∥2+∥ϑn∥2+∥ϑn-1∥2+∥ϑn-2∥2).
Summing fromn=2,…,J, the resulting inequality becomes
(58)γΔt∑n=2J‍∥κn-1/2∥2+(1-CΔt)∥ϑJ∥2≤∥ϑ1∥2+CΔt∑n=2J‍(∥R1n-1/2∥2+∥R2n-1/2∥2+∥πn-1/2∥2)+CΔt∑n=0J-1‍(∥τn∥2+∥ϑn∥2)+C∫0tJ‍∥τt∥2ds.
For sufficiently smallΔt, using Gronwall lemma with Lemma 10, we get
(59)γΔt∑n=2J‍∥κn-1/2∥2+12∥ϑJ∥2≤∥ϑ1∥2+CΔt∑n=0J-1‍∥τn∥2+CΔt∑n=2J‍(∥R1n-1/2∥2+∥R2n-1/2∥2+∥πn-1/2∥2)+C∫0tJ‍∥τt∥2ds≤∥ϑ1∥2+C(∫0tJ‍∥τn∥L∞(L2)2+∥πn-1/2∥L∞(L2)2+∫0tJ‍∥τt∥2ds)+CΔt4∥uttt∥L∞(L2)2.
Use Lemmas3 and 4, (59) with the triangle inequality to accomplish the proof of Theorem 12.Theorem 13.
Withuh0=𝒬hu0, assume that the solution's regular properties for system (6a), (6b), and (6c) satisfy ut∈L2(H2), uttt∈L∞(L2), u,w∈L∞(H2), χ∈L∞(H1), χt∈L2(H1). Then there exists a positive constant C independent of h and  Δt such that (60a)∥u(tJ)-uhJ∥1≤C[h∥|·|∥+Δt2∥uttt∥L∞(L2)],(60b)(Δt∑n=1J‍∥w(tn-1/2)-whn-1/2∥j2)1/2≤C[h2-j∥|·|∥+Δt2∥uttt∥L∞(L2)],j=0,1,(60c)∥χ(tn-1/2)-χhn-1/2∥≤C[h∥|·|∥+Δt2∥uttt∥L∞(L2)].Proof.
Choosevh=κn-1/2, gh=(ϑn-ϑn-1)/Δt, and zh=(∇ϑn-∇ϑn-1)/Δt in (50a), (50b), and (50c) to obtain (61a)(ϑn-ϑn-1Δt,κn-1/2)+γ∥∇κn-1/2∥2+∥κn-1/2∥2+(f(u^^n-1/2)-f(u^^hn-1/2),κn-1/2)=-(τn-τn-1Δt,κn-1/2)-(πn-1/2,κn-1/2)+(R1n-1/2,κn-1/2)+(R2n-1/2,κn-1/2),(61b)(ζn-1/2,∇ϑn-∇ϑn-1Δt)-(κn-1/2,ϑn-ϑn-1Δt)=(πn-1/2,ϑn-ϑn-1Δt),(61c)-(ζn-1/2,∇ϑn-∇ϑn-1Δt)+∥∇ϑn∥2-∥∇ϑn-1∥22Δt=0.
We add the above three equations of (61a), (61b), and (61c) to get
(62)∥∇ϑn∥2-∥∇ϑn-1∥22Δt+γ∥∇κn-1/2∥2+∥κn-1/2∥2=-(f(u^^n-1/2)-f(u^^hn-1/2),κn-1/2)-(τn-τn-1Δt,κn-1/2)-(πn-1/2,κn-1/2)+(R1n-1/2,κn-1/2)+(R2n-1/2,κn-1/2)+(πn-1/2,ϑn-ϑn-1Δt)=-(f(u^^n-1/2)-f(u^^hn-1/2),κn-1/2)-(τn-τn-1Δt,κn-1/2)-(πn-1/2,κn-1/2)+(R1n-1/2,κn-1/2)+(R2n-1/2,κn-1/2)+(πn,ϑn)-(πn-1,ϑn-1)Δt-(ϑn-1/2,πn-πn-1Δt).
For (62), we employ Cauchy-Schwarz inequality as well as Young inequality to obtain
(63)∥∇ϑn∥2-∥∇ϑn-1∥22Δt+γ∥∇κn-1/2∥2+12∥κn-1/2∥2≤C(∥f(u^^n-1/2)-f(u^^hn-1/2)∥2+∥τn-τn-1Δt∥2+∥πn-1/2∥2+∥R1n-1/2∥2+∥R2n-1/2∥2+∥ϑn-1/2∥2+∥πn-πn-1Δt∥2)+(πn,ϑn)-(πn-1,ϑn-1)Δt.
Multiply (63) by 2Δt  and then sum from n=2,…,J to get
(64)∥∇ϑJ∥2+2γΔt∑n=2J‍∥∇κn-1/2∥2+Δt∑n=2J‍∥κn-1/2∥2≤∥∇ϑ1∥2+CΔt∑n=2J‍(∥πn-πn-1Δt∥2∥f(u^^n-1/2)-f(u^^hn-1/2)∥2+∥τn-τn-1Δt∥2+∥πn-1/2∥2+∥R1n-1/2∥2+∥R2n-1/2∥2+∥ϑn-1/2∥2+∥πn-πn-1Δt∥2)+(πJ,ϑJ)-(π1,ϑ1).
Substituting the following inequality(65)∥πn-πn-1Δt∥2≤1Δt∫tn-1tn‍∥πt(s)∥2ds,
and (55) into (64), we obtain
(66)∥∇ϑJ∥2+2γΔt∑n=2J‍∥∇κn-1/2∥2+Δt∑n=2J‍∥κn-1/2∥2≤∥ϑ1∥12+∥πJ∥2+CΔt∑n=2J‍(∥ϑn-1/2∥2∥τn-1∥2+∥τn-2∥2+∥ϑn-1∥2+∥ϑn-2∥2+∥πn-1/2∥2+∥R1n-1/2∥2+∥R2n-1/2∥2+∥ϑn-1/2∥2)+∥ϑJ∥2.
Substitute (59) into (66) to obtain
(67)∥∇ϑJ∥2+2γΔt∑n=2J‍∥∇κn-1/2∥2+Δt∑n=2J‍∥κn-1/2∥2≤∥ϑ1∥12+CΔt∑n=2J‍(∥τn-1∥2+∥τn-2∥2+∥πn-1/2∥2+∥R1n-1/2∥2+∥R2n-1/2∥2)+∥πJ∥2+C(∥τn∥L∞(L2)2+∫0tJ‍∥τt∥2ds+Δt4∥uttt∥L∞(L2)+∥πn-1/2∥L∞(L2)2∫0tJ‍).
Takingzh=ζn-1/2 in (50c), we have
(68)∥ζn-1/2∥2≤C∥∇ϑn-1/2∥2≤C(∥ϑn∥2+∥ϑn-1∥2).
Combining Lemmas3–10, (67), and (68) with the triangle inequality, we accomplish the proof.


## 4.1. Linearized Crank-Nicolson Mixed Scheme

In the following discussion, we will derive the fully discrete a priori error estimates based on a linearized Crank-Nicolson method. Let0=t0<t1<t2<⋯<tM=T be a given partition of the time interval [0,T] with step length Δt=T/M and nodes tn=nΔt, for some positive integer M. For a smooth function ϕ on [0,T], define ψn=ϕ(tn) and ψn-1/2=(ψn+ψn-1)/2.Equations (6a), (6b), and (6c) can be written as the following formulation at time t=tn-1/2: (44a)(ut(tn-1/2),v)+γ(∇w(tn-1/2),∇v)+(w(tn-1/2),v)+(f(u(tn-1/2)),v)=0,∀v∈H01,(44b)(χ(tn-1/2),∇g)-(w(tn-1/2),g)=0,∀g∈H01m,(44c)(χ(tn-1/2),z)-(∇u(tn-1/2),z)=0,∀z∈L2.Then, the following equivalent formulation for (44a), (44b), and (44c) is as follows: (45a)(un-un-1Δt,v)+γ(∇wn-1/2,∇v)+(wn-1/2,v)+(f(u^^n-1/2),v)=(R1n-1/2,v)+(R2n-1/2,v),∀v∈H01,(45b)(χn-1/2,∇g)-(wn-1/2,g)=0,∀g∈H01,(45c)(χn-1/2,z)-(∇un-1/2,z)=0,∀z∈L2,where(46)u^^n-1/2=32un-1-12un-2,R1n-1/2=un-un-1Δt-ut(tn-1/2),R2n-1/2=f(u^^n-1/2)-f(u(tn-1/2)).Now a linearized C-N fully discrete procedure is to find(uhn,whn,χhn)∈Hh×Hh×Lh,(n=0,1,…,M) such that (47a)(uhn-uhn-1Δt,vh)+γ(∇whn-1/2,∇vh)+(whn-1/2,vh)+(f(u^^hn-1/2),vh)=0,∀vh∈Hh,(47b)(χhn-1/2,∇gh)-(whn-1/2,gh)=0,∀gh∈Hh,(47c)(χhn-1/2,zh)-(∇uhn-1/2,zh)=0,∀zh∈Lh.Remark 9.
We can find that the system (47a), (47b), and (47c) is a linear scheme by a linearized term
(48)f(u^^hn-1/2)=(32uhn-1-12uhn-2)3-(32uhn-1-12uhn-2).In the following subsection, we derive a priori error estimates based on a linearized Crank-Nicolson fully discrete scheme.


## 4.2. A Priori Error Estimates for Fully Discrete Scheme

In order to derive the linearized C-N fully discrete error estimates, we now write the errors as follows:(49)w(tn)-whn=w(tn)-𝒮hwn+𝒮hwn-whn=πn+κn;u(tn)-uhn=u(tn)-𝒬hun+𝒬hun-uhn=τn+ϑn;χ(tn)-χhn=χ(tn)-𝒬hχn+𝒬hχn-χhn=λn+ζn.Subtracting (47a), (47b), and (47c) from (45a), (45b), and (45c) and using (19) and (20a) and (20b) at t=tn-1/2, we have the following error equations (50a)(ϑn-ϑn-1Δt,vh)+γ(∇κn-1/2,∇vh)+(κn-1/2,vh)+(f(u^^n-1/2)-f(u^^hn-1/2),vh)=-(τn-τn-1Δt,vh)-(πn-1/2,vh)+(R1n-1/2,vh)+(R2n-1/2,vh),∀vh∈Hh,(50b)(ζn-1/2,∇gh)-(κn-1/2,gh)=(πn-1/2,gh),∀gh∈Hh,(50c)(ζn-1/2,zh)-(∇ϑn-1/2,zh)=0,∀zh∈Lh.In order to get the fully discrete error estimates, we introduce the following lemma.Lemma 10.
ForR1n-1/2 and R2n-1/2, the following estimates hold:
(51)∥R1n-1/2∥≤CΔt2∥uttt∥L∞(L2),∥R2n-1/2∥≤C(u)Δt2∥uttt∥L∞(L2).Proof.
Using the Taylor expansion, we can obtain easily the conclusion for Lemma10.Remark 11.
From the estimate for∥R2n-1/2∥ in Lemma 10, we know that the convergence rate in time direction for the system (47a), (47b), and (47c) is order 2, so the system (47a), (47b), and (47c) is called a linearized Crank-Nicolson scheme based on the linearized term f(u^^hn-1/2).In the following discussion, we will derive some fully discrete a priori error estimates.Theorem 12.
Withuh0=𝒬hu0, assume that the solution's regular properties for system (6a), (6b), and (6c) satisfy ut∈L2(H2), uttt∈L∞(L2), u,w∈L∞(H2), χ∈L∞(H1), χt∈L2(H1). Then there exists a positive constant C independent of h and Δt such that
(52)∥u(tJ)-uhJ∥≤C[h2∥|·|∥+Δt2∥uttt∥L∞(L2)],∥|·|∥≜∥u∥L∞(H2)+∥ut∥L2(H2)+∥w∥L∞(H2)+∥χ∥L∞(H1)+∥χt∥L2(H1).Proof.
Settingvh=ϑn-1/2, gh=κn-1/2, and zh=∇κn-1/2 in (50a), (50b), and (50c), respectively, we get (53a)∥ϑn∥2-∥ϑn-1∥22Δt+γ(∇κn-1/2,∇ϑn-1/2)+(κn-1/2,ϑn-1/2)+(f(u^^n-1/2)-f(u^^hn-1/2),ϑn-1/2)=-(τn-τn-1Δt,ϑn-1/2)-(πn-1/2,ϑn-1/2)+(R1n-1/2,ϑn-1/2)+(R2n-1/2,ϑn-1/2),(53b)γ∥κn-1/2∥2-γ(ζn-1/2,∇κn-1/2)=-γ(πn-1/2,κn-1/2),(53c)γ(ζn-1/2,∇κn-1/2)-γ(∇ϑn-1/2,∇κn-1/2)=0.
Adding the three equations of (53a), (53b), and (53c) and using Cauchy-Schwarz inequality and Young inequality, we have
(54)γ∥κn-1/2∥2+12Δt(∥ϑn∥2-∥ϑn-1∥2)=-(κn-1/2,ϑn-1/2)-(f(u^^n-1/2)-f(u^^hn-1/2),ϑn-1/2)-(τn-τn-1Δt,ϑn-1/2)-(πn-1/2,ϑn-1/2)+(R1n-1/2,ϑn-1/2)+(R2n-1/2,ϑn-1/2)-γ(πn-1/2,κn-1/2)≤(∥τn-τn-1Δt∥+∥R1n-1/2∥+∥R2n-1/2∥)∥ϑn-1/2∥+∥κn-1/2∥∥ϑn-1/2∥+γ∥πn-1/2∥∥κn-1/2∥+(∥f(u^^n-1/2)-f(u^^hn-1/2)∥+∥πn-1/2∥)∥ϑn-1/2∥≤C(∥τn-τn-1Δt∥2+∥R1n-1/2∥2+∥R2n-1/2∥2+∥πn-1/2∥2)+γ2∥κn-1/2∥2+C∥f(u^^n-1/2)-f(u^^hn-1/2)∥2+C∥ϑn-1/2∥2.
In order to accomplish our process, we have to estimate the nonlinear term∥f(u^^n-1/2)-f(u^^hn-1/2)∥2. Using Young inequality, we have
(55)∥f(u^^n-1/2)-f(u^^hn-1/2)∥2≤2∥(u^^n-1/2)3-(u^^hn-1/2)3∥2+2∥u^^n-1/2-u^^hn-1/2∥2=2∥[(12)232(un-1-uhn-1)-12(un-2-uhn-2)]×[(12)2(32un-1-12un-2)2+(32uhn-1-12uhn-2)2+(32un-1-12un-2)(32uhn-1-12uhn-2)(12)2]∥2+2∥32(un-1-uhn-1)-12(un-2-uhn-2)∥2≤2∥32(ϑn-1+τn-1)-12(ϑn-2+τn-2)∥2×[(12)2∥(32un-1-12un-2)2+(32uhn-1-12uhn-2)2+(32un-1-12un-2)(32uhn-1-12uhn-2)(12)2∥L∞2+1(12)2]=(24Cu2+2)∥32(ϑn-1+τn-1)-12(ϑn-2+τn-2)∥2≤C(∥τn-1∥2+∥τn-2∥2+∥ϑn-1∥2+∥ϑn-2∥2).
Note that(56)∥τn-τn-1Δt∥2≤1Δt∫tn-1tn‍∥τt(s)∥2ds.
Substitute (55) and (56) into (54) to obtain
(57)γ2∥κn-1/2∥2+12Δt(∥ϑn∥2-∥ϑn-1∥2)≤C(1Δt∫tn-1tn‍∥τt(s)∥2ds+∥R1n-1/2∥2+∥R2n-1/2∥2+∥πn-1/2∥2∫tn-1tn‍)+C(∥τn-1∥2+∥τn-2∥2+∥ϑn∥2+∥ϑn-1∥2+∥ϑn-2∥2).
Summing fromn=2,…,J, the resulting inequality becomes
(58)γΔt∑n=2J‍∥κn-1/2∥2+(1-CΔt)∥ϑJ∥2≤∥ϑ1∥2+CΔt∑n=2J‍(∥R1n-1/2∥2+∥R2n-1/2∥2+∥πn-1/2∥2)+CΔt∑n=0J-1‍(∥τn∥2+∥ϑn∥2)+C∫0tJ‍∥τt∥2ds.
For sufficiently smallΔt, using Gronwall lemma with Lemma 10, we get
(59)γΔt∑n=2J‍∥κn-1/2∥2+12∥ϑJ∥2≤∥ϑ1∥2+CΔt∑n=0J-1‍∥τn∥2+CΔt∑n=2J‍(∥R1n-1/2∥2+∥R2n-1/2∥2+∥πn-1/2∥2)+C∫0tJ‍∥τt∥2ds≤∥ϑ1∥2+C(∫0tJ‍∥τn∥L∞(L2)2+∥πn-1/2∥L∞(L2)2+∫0tJ‍∥τt∥2ds)+CΔt4∥uttt∥L∞(L2)2.
Use Lemmas3 and 4, (59) with the triangle inequality to accomplish the proof of Theorem 12.Theorem 13.
Withuh0=𝒬hu0, assume that the solution's regular properties for system (6a), (6b), and (6c) satisfy ut∈L2(H2), uttt∈L∞(L2), u,w∈L∞(H2), χ∈L∞(H1), χt∈L2(H1). Then there exists a positive constant C independent of h and  Δt such that (60a)∥u(tJ)-uhJ∥1≤C[h∥|·|∥+Δt2∥uttt∥L∞(L2)],(60b)(Δt∑n=1J‍∥w(tn-1/2)-whn-1/2∥j2)1/2≤C[h2-j∥|·|∥+Δt2∥uttt∥L∞(L2)],j=0,1,(60c)∥χ(tn-1/2)-χhn-1/2∥≤C[h∥|·|∥+Δt2∥uttt∥L∞(L2)].Proof.
Choosevh=κn-1/2, gh=(ϑn-ϑn-1)/Δt, and zh=(∇ϑn-∇ϑn-1)/Δt in (50a), (50b), and (50c) to obtain (61a)(ϑn-ϑn-1Δt,κn-1/2)+γ∥∇κn-1/2∥2+∥κn-1/2∥2+(f(u^^n-1/2)-f(u^^hn-1/2),κn-1/2)=-(τn-τn-1Δt,κn-1/2)-(πn-1/2,κn-1/2)+(R1n-1/2,κn-1/2)+(R2n-1/2,κn-1/2),(61b)(ζn-1/2,∇ϑn-∇ϑn-1Δt)-(κn-1/2,ϑn-ϑn-1Δt)=(πn-1/2,ϑn-ϑn-1Δt),(61c)-(ζn-1/2,∇ϑn-∇ϑn-1Δt)+∥∇ϑn∥2-∥∇ϑn-1∥22Δt=0.
We add the above three equations of (61a), (61b), and (61c) to get
(62)∥∇ϑn∥2-∥∇ϑn-1∥22Δt+γ∥∇κn-1/2∥2+∥κn-1/2∥2=-(f(u^^n-1/2)-f(u^^hn-1/2),κn-1/2)-(τn-τn-1Δt,κn-1/2)-(πn-1/2,κn-1/2)+(R1n-1/2,κn-1/2)+(R2n-1/2,κn-1/2)+(πn-1/2,ϑn-ϑn-1Δt)=-(f(u^^n-1/2)-f(u^^hn-1/2),κn-1/2)-(τn-τn-1Δt,κn-1/2)-(πn-1/2,κn-1/2)+(R1n-1/2,κn-1/2)+(R2n-1/2,κn-1/2)+(πn,ϑn)-(πn-1,ϑn-1)Δt-(ϑn-1/2,πn-πn-1Δt).
For (62), we employ Cauchy-Schwarz inequality as well as Young inequality to obtain
(63)∥∇ϑn∥2-∥∇ϑn-1∥22Δt+γ∥∇κn-1/2∥2+12∥κn-1/2∥2≤C(∥f(u^^n-1/2)-f(u^^hn-1/2)∥2+∥τn-τn-1Δt∥2+∥πn-1/2∥2+∥R1n-1/2∥2+∥R2n-1/2∥2+∥ϑn-1/2∥2+∥πn-πn-1Δt∥2)+(πn,ϑn)-(πn-1,ϑn-1)Δt.
Multiply (63) by 2Δt  and then sum from n=2,…,J to get
(64)∥∇ϑJ∥2+2γΔt∑n=2J‍∥∇κn-1/2∥2+Δt∑n=2J‍∥κn-1/2∥2≤∥∇ϑ1∥2+CΔt∑n=2J‍(∥πn-πn-1Δt∥2∥f(u^^n-1/2)-f(u^^hn-1/2)∥2+∥τn-τn-1Δt∥2+∥πn-1/2∥2+∥R1n-1/2∥2+∥R2n-1/2∥2+∥ϑn-1/2∥2+∥πn-πn-1Δt∥2)+(πJ,ϑJ)-(π1,ϑ1).
Substituting the following inequality(65)∥πn-πn-1Δt∥2≤1Δt∫tn-1tn‍∥πt(s)∥2ds,
and (55) into (64), we obtain
(66)∥∇ϑJ∥2+2γΔt∑n=2J‍∥∇κn-1/2∥2+Δt∑n=2J‍∥κn-1/2∥2≤∥ϑ1∥12+∥πJ∥2+CΔt∑n=2J‍(∥ϑn-1/2∥2∥τn-1∥2+∥τn-2∥2+∥ϑn-1∥2+∥ϑn-2∥2+∥πn-1/2∥2+∥R1n-1/2∥2+∥R2n-1/2∥2+∥ϑn-1/2∥2)+∥ϑJ∥2.
Substitute (59) into (66) to obtain
(67)∥∇ϑJ∥2+2γΔt∑n=2J‍∥∇κn-1/2∥2+Δt∑n=2J‍∥κn-1/2∥2≤∥ϑ1∥12+CΔt∑n=2J‍(∥τn-1∥2+∥τn-2∥2+∥πn-1/2∥2+∥R1n-1/2∥2+∥R2n-1/2∥2)+∥πJ∥2+C(∥τn∥L∞(L2)2+∫0tJ‍∥τt∥2ds+Δt4∥uttt∥L∞(L2)+∥πn-1/2∥L∞(L2)2∫0tJ‍).
Takingzh=ζn-1/2 in (50c), we have
(68)∥ζn-1/2∥2≤C∥∇ϑn-1/2∥2≤C(∥ϑn∥2+∥ϑn-1∥2).
Combining Lemmas3–10, (67), and (68) with the triangle inequality, we accomplish the proof.


## 5. Some Concluding Remarks and Extensions

In [29, 30], authors proposed a new mixed finite element method, which has been applied to solve second-order evolution equations, such as Sobolev equations [31], parabolic equations [32, 33], and reaction-convection-diffusion problems [34, 35]. In this paper, we apply the new mixed finite element scheme [29, 30] to solve the extended Fisher-Kolmogorov (EFK) equation (fourth-order nonlinear reaction diffusion equation). Compared to the classical mixed methods, the weaker square integrable L2 space, which takes the place of the complex H(div;Ω), is used in our method. We derive some a priori bounds for the solution and a priori error estimates for semidiscrete scheme. What's more, we obtain a priori error estimates for fully discrete scheme by a linearized Crank-Nicolson method.If we takeγ=0 in (1), we can get the following second-order nonlinear reaction-diffusion equation:
(69)ut-Δu+f(u)=0,(x,t)∈Ω×J,with initial condition(70)u(x,0)=u0(x),x∈Ω,and Dirichlet boundary conditions(71)u(x,t)=0,(x,t)∈∂Ω×J-,whereΩ is a bounded convex polygonal domain in Rd, d≤2 with boundary ∂Ω, and J=(0,T] is the time interval with 0<T<∞. u0(x) is given function, f(u)=u3-u.Using a similar method as the one in this paper, we can get the following weak formulation for system (69)–(71): (72a)(ut,v)+(λ,∇v)+(f(u),v)=0,∀v∈H01,(72b)(λ,z)-(∇u,z)=0,∀z∈L2,whereλ=∇u. Based on the mixed weak formulation (72a) and (72b), we can get the similar theoretical analysis as our method in this paper.In the future work, we will apply the new mixed scheme to solve fourth-order linear/nonlinear wave equations [24, 25, 38]. At the same time, we will study the large time-stepping method based on the new mixed element scheme for the Cahn-Hilliard equation [19, 20].



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