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

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.

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] 𝑢_𝑡+𝛾Δ²𝑢−Δ𝑢+𝑓(𝑢)=0,(𝐱,𝑡)∈Ω×𝐽, (1)

with initial condition 𝑢(𝐱,0)=𝑢₀(𝐱),𝐱∈Ω, (2)

and Dirichlet boundary conditions 𝑢(𝐱,𝑡)=Δ𝑢(𝐱,𝑡)=0,(𝐱,𝑡)∈𝜕Ω×𝐽, (3)

where Ω is a bounded convex polygonal domain in 𝑅^𝑑, 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 term 𝑤=−Δ𝑢 to get the following two lower order equations: 𝑢_𝑡(𝑢),()()−𝛾Δ𝑤+𝑤=−𝑓𝐱,𝑡∈Ω×𝐽,−Δ𝑢=𝑤,𝐱,𝑡∈Ω×𝐽. (4a)(4b)

Then, we introduce another auxiliary variable 𝝌=∇𝑢 in (4b) to obtain the following lower-order system: 𝑢_𝑡−𝛾Δ𝑤+𝑤=−𝑓(𝑢),−∇⋅𝝌=𝑤,𝝌−∇𝑢=0. (5a)(5b)(5c)

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 Section 2, 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 𝒬_ℎ and a projection operator 𝒮_ℎ 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 𝐻¹₀≜𝐻¹₀(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]↦𝐻¹₀×𝐻¹₀×𝐋² such that 𝑢_𝑡,𝑣+𝛾(∇𝑤,∇𝑣)+(𝑤,𝑣)+𝑓(𝑢),𝑣=0,∀𝑣∈𝐻¹₀,𝝌,∇𝑔=𝑤,𝑔,∀𝑔∈𝐻¹₀,(𝝌,𝐳)−(∇𝑢,𝐳)=0,∀𝐳∈𝐋². (6a)(6b)(6c)

Let 𝐻_ℎ and 𝐋_ℎ, respectively, be finite dimensional subspaces of 𝐻¹₀ and 𝐋² defined by the finite element pair 𝑃₁−𝑃²₀ [29, 30] 𝐻_ℎ=𝑣_ℎ∈𝐶⁰(Ω)∩𝐻¹₀∣𝑣_ℎ∈𝑃₁(𝐾),∀𝐾∈𝒦_ℎ,𝐋_ℎ=𝐰_ℎ=𝑤_1ℎ,𝑤_2ℎ∈𝐋²∣𝑤_1ℎ,𝑤_2ℎ∈𝑃₀(𝐾),∀𝐾∈𝒦_ℎ. (7)

Then, the semidiscrete mixed scheme for (6a), (6b), and (6c) is to determine {𝑢_ℎ,𝑤_ℎ,𝝌_ℎ}∶[0,𝑇]↦𝐻_ℎ×𝐻_ℎ×𝐋_ℎ such that 𝑢_ℎ𝑡,𝑣_ℎ+𝛾∇𝑤_ℎ,∇𝑣_ℎ+𝑤_ℎ,𝑣_ℎ+𝑓𝑢_ℎ,𝑣_ℎ=0,∀𝑣_ℎ∈𝐻_ℎ,𝝌_ℎ,∇𝑔_ℎ=𝑤_ℎ,𝑔_ℎ,∀𝑔_ℎ∈𝐻_ℎ,𝝌_ℎ,𝐳_ℎ−∇𝑢_ℎ,𝐳_ℎ=0,∀𝐳_ℎ∈𝐋_ℎ. (8a)(8b)(8c)

In the following discussion, we derive some a priori bounds for the finite element solution.

Theorem 1. Let {𝑢_ℎ,𝑤_ℎ,𝝌_ℎ} be the solution for system (8a), (8b), and (8c) then there exists a constant 𝐶>0 such that, for 1≤𝑝<+∞𝑢_ℎ𝐿𝑝+𝑢_ℎ1+𝑤_ℎ+𝝌_ℎ+^𝑡₀𝑢_ℎ𝑡²𝑑𝑠^1/2≤𝐶𝛾,𝑢_0ℎ,𝑤_0ℎ. (9)

Proof. Differentiating (8b) and (8c) with respect to 𝑡, we obtain 𝑢_ℎ𝑡,𝑣_ℎ+𝛾∇𝑤_ℎ,∇𝑣_ℎ+𝑤_ℎ,𝑣_ℎ+𝑓𝑢_ℎ,𝑣_ℎ=0,∀𝑣_ℎ∈𝐻_ℎ,−𝝌_ℎ𝑡,∇𝑔_ℎ+𝑤_ℎ𝑡,𝑔_ℎ,∀𝑔_ℎ∈𝐻_ℎ,𝝌_ℎ𝑡,𝐳_ℎ−∇𝑢_ℎ𝑡,𝐳_ℎ=0,∀𝐳_ℎ∈𝐋_ℎ. (10a)(10b)(10c)
Take 𝑣_ℎ=𝑢_ℎ𝑡, 𝑔_ℎ=𝑤_ℎ, 𝐳_ℎ=∇𝑤_ℎ in (10a), (10b), and (10c) to obtain ‖𝑢_ℎ𝑡‖²+𝛾∇𝑤_ℎ,∇𝑢_ℎ𝑡+𝑤_ℎ,𝑢_ℎ𝑡+𝑓𝑢_ℎ,𝑢_ℎ𝑡𝝌=0,−𝛾_ℎ𝑡,∇𝑤_ℎ+𝛾2𝑑𝑤𝑑𝑡^ℎ2𝛾𝝌=0,_ℎ𝑡,∇𝑤_ℎ−𝛾∇𝑢_ℎ𝑡,∇𝑤_ℎ=0. (11a)(11b)(11c)
We define 𝐹(𝜎)≜(1/4)(1−𝜎²)² and add the three equations for system (11a), (11b), and (11c) to get 𝑑𝛾𝑑𝑡2𝑤_ℎ²+𝐹𝑢_ℎ,1+𝑢_ℎ𝑡²=−𝑤_ℎ,𝑢_ℎ𝑡≤12𝑤_ℎ²+12𝑢_ℎt². (12)
Integrate (12) with respect to time from 0 to 𝑡 and use Gronwall lemma to have 𝛾2𝑤_ℎ²+𝐹𝑢_ℎ,1+^𝑡₀𝑢_ℎ𝑡²𝛾𝑤𝑑𝑠≤𝐶_ℎ(0)²+𝐹𝑢_ℎ(0),1. (13)
We note that 𝐹(𝑢_ℎ)≥0 to obtain 𝑢_ℎ+𝑤_ℎ≤𝐶𝛾,𝑢_0ℎ,𝑤_0ℎ. (14)
Take 𝑔_ℎ=𝑢_ℎ in (8b) and 𝐳_ℎ=∇𝑢_ℎ in (8c) to obtain ∇𝑢_ℎ≤𝐶𝑢_ℎ+𝑤_ℎ. (15)
Substitute (14) into (15) to obtain ∇𝑢_ℎ≤𝐶𝛾,𝑢_0ℎ,𝑤_0ℎ. (16)
Choose 𝐳_ℎ=𝝌_ℎ in (8c) and use (16) to obtain 𝝌_ℎ≤∇𝑢_ℎ≤𝐶𝛾,𝑢_0ℎ,𝑤_0ℎ. (17)
For Ω⊂𝑅², we use Sobolev imbedding theorem to get 𝑢_ℎ𝐿𝑝≤𝑢_ℎ1≤𝐶𝛾,𝑢_0ℎ,𝑤_0ℎ. (18)
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 (𝐻_ℎ,𝐋_ℎ) 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 operator 𝒮_ℎ and a new mixed elliptic projection operator 𝒬_ℎ associated with the coupled equations.

Lemma 3. Let 𝒮_ℎ∶[0,𝑇]→𝐻¹₀→𝐻_ℎ be satisfied by the following relation: ∇𝑤−𝒮_ℎ𝑤,∇𝑣_ℎ=0,∀𝑣_ℎ∈𝐻_ℎ,𝑤−𝒮_ℎ𝑤_𝐿2(Ω)+ℎ𝑤−𝒮_ℎ𝑤_𝐻1≤𝐶ℎ²‖𝑤‖_𝐻². (19)

Lemma 4. Let (𝒬_ℎ𝑢,𝒬_ℎ𝝌)∶[0,𝑇]→𝐻_ℎ×𝐋_ℎ be given by the following new mixed relations: −𝝌−𝒬_ℎ𝝌,∇𝑔_ℎ=0,∀𝑔_ℎ∈𝐻_ℎ,𝝌−𝒬_ℎ𝝌,𝐳_ℎ−∇𝑢−𝒬_ℎ𝑢,𝐳_ℎ=0,∀𝐳_ℎ∈𝐋_ℎ; (20a)(20b)
there exists a constant 𝐶>0 independent of ℎ such that 𝑢−𝒬_ℎ𝑢+ℎ𝝌−𝒬_ℎ𝝌+𝑢−𝒬_ℎ𝑢₁≤𝐶ℎ²‖𝑢‖_𝐻²+‖𝝌‖_𝐇¹,𝑢_𝑡−𝒬_ℎ𝑢_𝑡≤𝐶ℎ²𝑢_𝑡𝐻2+𝝌_𝑡𝐇1, (21)

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,∞: 𝒬_ℎ𝑢_{𝑊𝑗,𝑝}≤‖𝜏‖_𝑊^𝑗,𝑝+‖𝑢‖_𝑊^𝑗,𝑝≤𝐶_𝑢. (22)

Based on Lemmas 3 and 4, the following theorem for semidiscrete error estimates is obtained.

Theorem 5. With 𝑢_ℎ(0)=𝒬_ℎ𝑢₀, assume that the solution's regular properties for system (6a), (6b), and (6c) satisfy 𝑤,𝑢,𝑢_𝑡∈𝐿^∞(𝐻²), 𝝌,𝝌_𝑡∈𝐿^∞(𝐇¹). Then there exists a positive constant 𝐶>0 independent of ℎ such that 𝑢−𝑢_ℎ+𝑤−𝑤_ℎ+ℎ𝑢−𝑢_ℎ1+𝝌−𝝌_ℎ≤𝐶ℎ²‖𝑢‖_𝐿^∞_(𝐻²₎+‖𝝌‖_𝐿^∞_(𝐇¹₎+^𝑡₀‖𝑢‖²_𝐻2+𝑢_𝑡²_𝐻2+‖𝝌‖²_𝐇1+𝝌_𝑡²_𝐇1+‖𝑤‖²_𝐻2𝑑𝑠^1/2. (23)

Proof. To get a priori error estimates, we decompose the errors as 𝑤−𝑤_ℎ=𝑤−𝒮_ℎ𝑤+𝒮_ℎ𝑤−𝑤_ℎ=𝜋+𝜅;𝑢−𝑢_ℎ=𝑢−𝒬_ℎ𝑢+𝒬_ℎ𝑢−𝑢_ℎ=𝜏+𝜗;𝝌−𝝌_ℎ=𝝌−𝒬_ℎ𝝌+𝒬_ℎ𝝌−𝝌_ℎ=𝝀+𝜻. (24)
Combining (6a)–(8c), (19) and (20a) and (20b), we can get the error equations 𝜗_𝑡,𝑣_ℎ+𝛾∇𝜅,∇𝑣_ℎ+𝜅,𝑣_ℎ+𝑓(𝑢)−𝑓𝑢_ℎ,𝑣_ℎ=−𝜏_𝑡,𝑣_ℎ−𝜋,𝑣_ℎ,∀𝑣_ℎ∈𝐻_ℎ,−𝜻,∇𝑔_ℎ+𝜅,𝑔_ℎ=−𝜋,𝑔_ℎ,∀𝑔_ℎ∈𝐻_ℎ,𝜻,𝐳_ℎ−∇𝜗,𝐳_ℎ=0,∀𝐳_ℎ∈𝐋_ℎ. (25a)(25b)(25c)
Differentiating (25b) and (25c) with respect to 𝑡, we get 𝜗_𝑡,𝑣_ℎ+𝛾∇𝜅,∇𝑣_ℎ+𝜅,𝑣_ℎ+𝑓(𝑢)−𝑓𝑢_ℎ,𝑣_ℎ=−𝜏_𝑡,𝑣_ℎ−𝜋,𝑣_ℎ,∀𝑣_ℎ∈𝐻_ℎ,−𝜻_𝑡,∇𝑔_ℎ+𝜅_𝑡,𝑔_ℎ=−𝜋_𝑡,𝑔_ℎ,∀𝑔_ℎ∈𝐻_ℎ,𝜻_𝑡,𝐳_ℎ−∇𝜗_𝑡,𝐳_ℎ=0,∀𝐳_ℎ∈𝐋_ℎ. (26a)(26b)(26c)
Choose 𝑣_ℎ=𝜗_𝑡, 𝑔_ℎ=𝜅, and 𝐳_ℎ=∇𝜅 in (26a), (26b), and (26c), respectively, to obtain 𝜗_𝑡²+𝛾∇𝜅,∇𝜗_𝑡+𝜅,𝜗_𝑡+𝑓(𝑢)−𝑓𝑢_ℎ,𝜗_𝑡=−𝜏_𝑡,𝜗_𝑡−𝜋,𝜗_𝑡,12𝑑‖𝜅‖𝑑𝑡²−𝜻_𝑡,∇𝜅=−𝜋_𝑡,𝜻,𝜅_𝑡,∇𝜅−∇𝜗_𝑡,∇𝜅=0. (27a)(27b)(27c)
Using (27a), (27b), and (27c) and Cauchy-Schwarz inequality and Young inequality, we have 𝜗_𝑡²+𝑓(𝑢)−𝑓𝑢_ℎ,𝜗_𝑡=−𝛾∇𝜅,∇𝜗_𝑡−𝜅,𝜗_𝑡−𝜏_𝑡,𝜗_𝑡−𝜋,𝜗_𝑡=−𝛾𝜻_𝑡,∇𝜅−𝜅,𝜗_𝑡−𝜏_𝑡,𝜗_𝑡−𝜋,𝜗_𝑡𝛾=−2𝑑‖𝜅‖𝑑𝑡²−𝜋_𝑡,𝜅−𝜅,𝜗_𝑡−𝜏_𝑡,𝜗_𝑡−𝜋,𝜗_𝑡𝛾≤−2𝑑‖𝜅‖𝑑𝑡²+𝐶𝜋_𝑡²+𝜏_𝑡²+‖𝜋‖²+‖𝜅‖²+14𝜗_𝑡². (28)
Using Cauchy-Schwarz inequality, Young inequality, Sobolev imbedding theorem, and inequality (22), we use the similar method as the one in [27] to obtain −𝑓(𝑢)−𝑓𝑢_ℎ,𝜗_𝑡=−𝑓(𝑢)−𝑓𝒬_ℎ𝑢,𝜗_𝑡−𝑓𝒬_ℎ𝑢−𝑓𝑢_ℎ,𝜗_𝑡=−𝑓(𝑢)−𝑓𝒬_ℎ𝑢,𝜗_𝑡−𝑓(𝜗),𝜗_𝑡−3𝑢_ℎ𝒬_ℎ𝑢𝜗,𝜗_𝑡≤𝜏𝑢²+𝑢𝒬_ℎ𝑢+𝒬_ℎ𝑢²−1,𝜗_𝑡−𝑑(𝐹(𝜗))𝑢𝑑𝑡,1−3_ℎ𝒬_ℎ𝑢𝜗,𝜗_𝑡≤𝐶‖𝜏‖²+‖𝜗‖²−𝑑(𝐹(𝜗))+1𝑑𝑡,14𝜗_𝑡². (29)
Substituting (29) into (28), we have 𝑑𝛾𝑑𝑡2‖𝜅‖²+(𝐹(𝜗)+1,1)2𝜗_𝑡²≤𝐶𝜏_𝑡²+‖𝜏‖²+‖𝜋‖²+𝜋_𝑡²+‖𝜗‖²+‖𝜅‖². (30)
Integrate (30) with respect to time from 0 to 𝑡 to obtain 𝛾2‖𝜅‖²+(𝐹(𝜗))+1,12^𝑡₀𝜗_𝑡²𝑑𝑠≤𝐶^𝑡₀𝜏_𝑡²+‖𝜏‖²+‖𝜋‖²+𝜋_𝑡²+‖𝜗‖²+‖𝜅‖²𝑑𝑠. (31)
Using Gronwall lemma, we have 𝛾2‖𝜅‖²+‖𝜗‖²+12^𝑡₀𝜗_𝑡²𝑑𝑠≤𝐶^𝑡₀𝜏_𝑡²+‖𝜏‖²+‖𝜋‖²+𝜋_𝑡²𝑑𝑠. (32)
Taking 𝑔_ℎ=𝜗 in (25b) and 𝐳_ℎ=∇𝜗 in (25c), we have ()−()=(),‖‖𝜻,∇𝜗𝜅,𝜗𝜋,𝜗∇𝜗²−(𝜻,∇𝜗)=0. (33a)(33b)
Add the two equations to get ‖∇𝜗‖²≤𝐶‖𝜅‖²+‖𝜗‖²+‖𝜋‖². (34)
Choose 𝐳_ℎ=𝜻 in (25c) to obtain ‖𝜻‖²≤‖∇𝜗‖²≤𝐶‖𝜅‖²+‖𝜗‖²+‖𝜋‖². (35)
Substitute (32) into (34) and (35) to get ‖𝜻‖²+‖∇𝜗‖²≤𝐶‖𝜋‖²+^𝑡₀𝜏_𝑡²+‖𝜏‖²+‖𝜋‖²+𝜋_𝑡²𝑑𝑠. (36)
Combining Lemmas 3 and 4, (32)–(36) with the triangle inequality, we obtain the error estimates for Theorem 5.

Remark 6. The conclusion for Theorem 5 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,𝑢_𝑡∈𝐿^∞(𝐻²) and 𝝌,𝝌_𝑡∈𝐿^∞(𝐇¹). Then there exists a positive constant 𝐶 independent of ℎ such that 𝑤−𝑤_ℎ1≤𝐶ℎ‖𝑢‖_𝐿^∞_(𝐻²₎+‖𝑤‖_𝐿^∞_(𝐻²₎+‖𝝌‖_𝐿^∞_(𝐇¹₎+^𝑡₀‖𝑢‖²_𝐻2+𝑢_𝑡²_𝐻2+‖𝝌‖²_𝐇1+𝝌_𝑡²_𝐇1+‖𝑤‖²_𝐻2𝑑𝑠^1/2. (37)

Proof. Choose 𝑣_ℎ=𝜅_𝑡, 𝑔_ℎ=𝜗_𝑡, and 𝐳_ℎ=𝜻_𝑡 in (26a), (26b), and (26c), respectively, to obtain𝜗_𝑡,𝜅_𝑡+𝛾2𝑑‖𝜅‖𝑑𝑡²¹+𝑓(𝑢)−𝑓𝑢_ℎ,𝜅_𝑡=−𝜏_𝑡,𝜅_𝑡−𝜋,𝜅_𝑡,𝜻_𝑡,∇𝜗_𝑡−𝜅_𝑡,𝜗_𝑡=𝜋_t,𝜗_𝑡,𝜻_𝑡²−∇𝜗_𝑡,𝜻_𝑡=0. (38a)(38b)(38c)
Adding the three equations of system (38a), (38b), and (38c), we have 𝛾2𝑑‖𝜅‖𝑑𝑡²¹+𝜻_𝑡²=−𝑓(𝑢)−𝑓𝑢_ℎ,𝜅_𝑡−𝜏_𝑡,𝜅_𝑡−𝜋,𝜅_𝑡+𝜋_𝑡,𝜗_𝑡𝑑=−𝑓(𝑢)𝑢𝑑𝑡−𝑓_ℎ,𝜅+𝑓^′(𝑢)𝑢_𝑡−𝑓^′𝑢_ℎ𝑢_ℎ𝑡−𝑑,𝜅𝜏𝑑𝑡^𝑡,𝜅+𝜏_𝑡𝑡−𝑑,𝜅()+𝜋𝑑𝑡𝜋,𝜅_𝑡,𝜅+𝜋_𝑡,𝜗_𝑡. (39)
Integrate (39) with respect to 𝑡 and use Cauchy-Schwarz inequality and Young inequality to get 𝛾2‖𝜅‖²₁+^𝑡₀𝜻_𝑡²𝑓(𝑢)𝑢𝑑𝑠=−−𝑓_ℎ+,𝜅_𝑡0𝑓^′(𝑢)𝑢_𝑡−𝑓^′𝑢_ℎ𝑢_ℎ𝑡−𝜏,𝜅𝑑𝑠_𝑡,𝜅+^𝑡₀𝜏_𝑡𝑡,𝜅𝑑𝑠−(𝜋,𝜅)+^𝑡₀𝜋_𝑡+,𝜅𝑑𝑠_𝑡0𝜋_𝑡,𝜗_𝑡𝑑𝑠≤−𝑓(𝑢)−𝑓𝑢_ℎ+,𝜅_𝑡0𝑓^′(𝑢)𝑢_𝑡−𝑓^′𝑢_ℎ𝑢_ℎ𝑡𝐶𝜏,𝜅𝑑𝑠,_𝑡2+‖𝜋‖²+^𝑡₀𝜏_𝑡𝑡²+𝜋_𝑡²+‖𝜅‖²+𝜗_𝑡²+𝛾𝑑𝑠8‖𝜅‖². (40)
Using the similar method to the estimate for (29), we obtain −𝑓(𝑢)−𝑓𝑢_ℎ𝑓(𝑢)𝑅,𝜅=−−𝑓_ℎ𝑢,𝜅−𝑓𝑅_ℎ𝑢−𝑓𝑢_ℎ≤𝜏𝑢,𝜅₂+𝑢𝑅_ℎ𝑢+𝑅_ℎ𝑢²+𝑓−1,𝜅_′𝑢_ℎ𝑅_ℎ𝑢−𝑢_ℎ=𝜏𝑢,𝜅₂+𝑢𝑅_ℎ𝑢+𝑅_ℎ𝑢²+3𝑢−1,𝜅_ℎ2‖𝜏‖−1𝜗,𝜅≤𝐶²+‖𝜗‖²+𝛾8‖𝜅‖²₁,_𝑡0𝑓^′(𝑢)𝑢_𝑡−𝑓^′𝑢_ℎ𝑢_ℎ𝑡,𝜅𝑑𝑠=^𝑡₀𝑓^′(𝑢)𝑢_𝑡−𝑢_ℎ𝑡+𝑓^′(𝑢)−𝑓^′𝑢_ℎ𝑢_ℎ𝑡=,𝜅𝑑𝑠_𝑡0𝑓^′(𝑢)𝜗_𝑡+𝜏_𝑡+𝑓^′′𝑢₁(𝜗+𝜏)𝑢_ℎ𝑡,𝜅𝑑𝑠≤𝐶^𝑡₀‖𝜏‖²+‖𝜗‖²+𝜏_𝑡²+𝜗_𝑡²+‖𝜅‖²₁𝑑𝑠. (41)
Substitute (41) into (40) to get 𝛾4‖𝜅‖²₁+^𝑡₀𝜻_𝑡²𝜏𝑑𝑠≤𝐶_𝑡²+‖𝜋‖²+^𝑡₀‖𝜏‖²+𝜏_𝑡²+𝜏_𝑡𝑡²+𝜋_𝑡²+‖𝜅‖²₁+‖𝜗‖²+𝜗_𝑡²𝑑𝑠. (42)
Substitute (32) into (42) and use Gronwall lemma to get 𝛾4‖𝜅‖²₁+^𝑡₀𝜻_𝑡²𝜏𝑑𝑠≤𝐶_𝑡²+‖𝜋‖²+^𝑡₀‖𝜏‖²+𝜏_𝑡²+𝜏_𝑡𝑡²+‖𝜋‖²+𝜋_𝑡²𝑑𝑠. (43)
Combining Lemmas 3 and 4, (43) with the triangle inequality, we obtain the error estimates for Theorem 7.

Remark 8. In Theorem 7, 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. Let 0=𝑡₀<𝑡₁<𝑡₂<⋯<𝑡_𝑀=𝑇 be a given partition of the time interval [0,𝑇] with step length Δ𝑡=𝑇/𝑀 and nodes 𝑡_𝑛=𝑛Δ𝑡, for some positive integer 𝑀. For a smooth function 𝜙 on [0,𝑇], define 𝜓^𝑛=𝜙(𝑡_𝑛) and 𝜓^𝑛−1/2=(𝜓^𝑛+𝜓^𝑛−1)/2.

Equations (6a), (6b), and (6c) can be written as the following formulation at time 𝑡=𝑡_𝑛−1/2: 𝑢_𝑡𝑡_𝑛−1/2,𝑣+𝛾∇𝑤𝑡_𝑛−1/2+𝑤𝑡,∇𝑣_𝑛−1/2,𝑣+𝑓𝑢𝑡_𝑛−1/2,𝑣=0,∀𝑣∈𝐻¹₀,𝝌𝑡_𝑛−1/2,∇𝑔−𝑤𝑡_𝑛−1/2,𝑔=0,∀𝑔∈𝐻¹₀𝝌𝑡𝑚,_𝑛−1/2,𝐳−∇𝑢𝑡_𝑛−1/2,𝐳=0,∀𝐳∈𝐋². (44a)(44b)(44c)

Then, the following equivalent formulation for (44a), (44b), and (44c) is as follows: 𝑢^𝑛−𝑢^𝑛−1Δ𝑡,𝑣+𝛾∇𝑤^𝑛−1/2+𝑤,∇𝑣_𝑛−1/2,𝑣+𝑓𝑢^𝑛−1/2=𝑅,𝑣_1𝑛−1/2,𝑣+𝑅₂^𝑛−1/2,𝑣,∀𝑣∈𝐻¹₀,𝝌^𝑛−1/2,∇𝑔−𝑤^𝑛−1/2,𝑔=0,∀𝑔∈𝐻¹₀,𝝌^𝑛−1/2,𝐳−∇𝑢^𝑛−1/2,𝐳=0,∀𝐳∈𝐋², (45a)(45b)(45c)

where 𝑢^𝑛−1/2=32𝑢^𝑛−1−12𝑢^𝑛−2,𝑅₁^𝑛−1/2=𝑢^𝑛−𝑢^𝑛−1Δ𝑡−𝑢_𝑡𝑡_𝑛−1/2,𝑅₂^𝑛−1/2=𝑓𝑢^𝑛−1/2−𝑓𝑢𝑡_𝑛−1/2. (46)

Now a linearized C-N fully discrete procedure is to find (𝑢^𝑛_ℎ,𝑤^𝑛_ℎ,𝝌^𝑛_ℎ)∈𝐻_ℎ×𝐻_ℎ×𝐋_ℎ,(𝑛=0,1,…,𝑀) such that 𝑢^𝑛_ℎ−𝑢_ℎ^𝑛−1Δ𝑡,𝑣_ℎ+𝛾∇𝑤_ℎ^𝑛−1/2,∇𝑣_ℎ+𝑤_ℎ^𝑛−1/2,𝑣_ℎ+𝑓𝑢_ℎ^𝑛−1/2,𝑣_ℎ=0,∀𝑣_ℎ∈𝐻_ℎ,𝝌_ℎ^𝑛−1/2,∇𝑔_ℎ−𝑤_ℎ^𝑛−1/2,𝑔_ℎ=0,∀𝑔_ℎ∈𝐻_ℎ,𝝌_ℎ^𝑛−1/2,𝐳_ℎ−∇𝑢_ℎ^𝑛−1/2,𝐳_ℎ=0,∀𝐳_ℎ∈𝐋_ℎ. (47a)(47b)(47c)

Remark 9. We can find that the system (47a), (47b), and (47c) is a linear scheme by a linearized term 𝑓𝑢_ℎ^𝑛−1/2=32𝑢_ℎ^𝑛−1−12𝑢_ℎ^𝑛−23−32𝑢_ℎ^𝑛−1−12𝑢_ℎ^𝑛−2. (48)

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: 𝑤𝑡_𝑛−𝑤^𝑛_ℎ=𝑤𝑡_𝑛−𝒮_ℎ𝑤^𝑛+𝒮_ℎ𝑤^𝑛−𝑤^𝑛_ℎ=𝜋^𝑛+𝜅^𝑛;𝑢𝑡_𝑛−𝑢^𝑛_ℎ=𝑢𝑡_𝑛−𝒬_ℎ𝑢^𝑛+𝒬_ℎ𝑢^𝑛−𝑢^𝑛_ℎ=𝜏^𝑛+𝜗^𝑛;𝝌𝑡_𝑛−𝝌^𝑛_ℎ=𝝌𝑡_𝑛−𝒬_ℎ𝝌^𝑛+𝒬_ℎ𝝌^𝑛−𝝌^𝑛_ℎ=𝝀^𝑛+𝜻^𝑛. (49)

Subtracting (47a), (47b), and (47c) from (45a), (45b), and (45c) and using (19) and (20a) and (20b) at 𝑡=𝑡_𝑛−1/2, we have the following error equations 𝜗^𝑛−𝜗^𝑛−1Δ𝑡,𝑣_ℎ+𝛾∇𝜅^𝑛−1/2,∇𝑣_ℎ+𝜅^𝑛−1/2,𝑣_ℎ+𝑓𝑢^𝑛−1/2−𝑓𝑢_ℎ^𝑛−1/2,𝑣_ℎ𝜏=−^𝑛−𝜏^𝑛−1Δ𝑡,𝑣_ℎ−𝜋^𝑛−1/2,𝑣_ℎ+𝑅₁^𝑛−1/2,𝑣_ℎ+𝑅₂^𝑛−1/2,𝑣_ℎ,∀𝑣_ℎ∈𝐻_ℎ,𝜻^𝑛−1/2,∇𝑔_ℎ−𝜅^𝑛−1/2,𝑔_ℎ=𝜋^𝑛−1/2,𝑔_ℎ,∀𝑔_ℎ∈𝐻_ℎ,𝜻^𝑛−1/2,𝐳_ℎ−∇𝜗^𝑛−1/2,𝐳_ℎ=0,∀𝐳_ℎ∈𝐋_ℎ. (50a)(50b)(50c)

In order to get the fully discrete error estimates, we introduce the following lemma.

Lemma 10. For 𝑅₁^𝑛−1/2 and 𝑅₂^𝑛−1/2, the following estimates hold: 𝑅₁^𝑛−1/2≤𝐶Δ𝑡²𝑢_{𝑡𝑡𝑡𝐿∞(𝐿2)},𝑅₂^𝑛−1/2≤𝐶(𝑢)Δ𝑡²𝑢_{𝑡𝑡𝑡𝐿∞(𝐿2)}. (51)

Proof. Using the Taylor expansion, we can obtain easily the conclusion for Lemma 10.

Remark 11. From the estimate for ‖𝑅₂^𝑛−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_ℎ^𝑛−1/2).

In the following discussion, we will derive some fully discrete a priori error estimates.

Theorem 12. With 𝑢⁰_ℎ=𝒬_ℎ𝑢⁰, assume that the solution's regular properties for system (6a), (6b), and (6c) satisfy 𝑢_𝑡∈𝐿²(𝐻²), 𝑢_𝑡𝑡𝑡∈𝐿^∞(𝐿²), 𝑢,𝑤∈𝐿^∞(𝐻²), 𝝌∈𝐿^∞(𝐇¹), 𝝌_𝑡∈𝐿²(𝐇¹). Then there exists a positive constant 𝐶 independent of ℎ and Δ𝑡 such that 𝑢𝑡_𝐽−𝑢^𝐽_ℎ≤𝐶ℎ²‖|⋅|‖+Δ𝑡²𝑢_{𝑡𝑡𝑡𝐿∞(𝐿2)}, (52)‖|⋅|‖≜‖𝑢‖_𝐿^∞_(𝐻²₎+‖𝑢_𝑡‖_𝐿2(𝐻2)+‖𝑤‖_𝐿^∞_(𝐻²₎+‖𝝌‖_𝐿^∞_(𝐇¹₎+‖𝝌_𝑡‖_𝐿2(𝐇¹₎.

Proof. Setting 𝑣_ℎ=𝜗^𝑛−1/2, 𝑔_ℎ=𝜅^𝑛−1/2, and 𝐳_ℎ=∇𝜅^𝑛−1/2 in (50a), (50b), and (50c), respectively, we get 𝜗^𝑛2−𝜗^𝑛−122Δ𝑡+𝛾∇𝜅^𝑛−1/2,∇𝜗^𝑛−1/2+𝜅^𝑛−1/2,𝜗^𝑛−1/2+𝑓𝑢^𝑛−1/2−𝑓𝑢_ℎ^𝑛−1/2,𝜗^𝑛−1/2𝜏=−^𝑛−𝜏^𝑛−1Δ𝑡,𝜗^𝑛−1/2−𝜋^𝑛−1/2,𝜗^𝑛−1/2+𝑅₁^𝑛−1/2,𝜗^𝑛−1/2+𝑅₂^𝑛−1/2,𝜗^𝑛−1/2,𝛾𝜅^𝑛−1/22−𝛾𝜻^𝑛−1/2,∇𝜅^𝑛−1/2=−𝛾𝜋^𝑛−1/2,𝜅^𝑛−1/2,𝛾𝜻^𝑛−1/2,∇𝜅^𝑛−1/2−𝛾∇𝜗^𝑛−1/2,∇𝜅^𝑛−1/2=0. (53a)(53b)(53c)
Adding the three equations of (53a), (53b), and (53c) and using Cauchy-Schwarz inequality and Young inequality, we have 𝛾𝜅^𝑛−1/22+1𝜗2Δ𝑡^𝑛2−𝜗^𝑛−12=−𝜅^𝑛−1/2,𝜗^𝑛−1/2−𝑓𝑢^𝑛−1/2−𝑓𝑢_ℎ^𝑛−1/2,𝜗^𝑛−1/2−𝜏^𝑛−𝜏^𝑛−1Δ𝑡,𝜗^𝑛−1/2−𝜋^𝑛−1/2,𝜗^𝑛−1/2+𝑅₁^𝑛−1/2,𝜗^𝑛−1/2+𝑅₂^𝑛−1/2,𝜗^𝑛−1/2−𝛾𝜋^𝑛−1/2,𝜅^𝑛−1/2≤𝜏^𝑛−𝜏^𝑛−1+𝑅Δ𝑡^1𝑛−1/2+𝑅₂^𝑛−1/2𝜗^𝑛−1/2+𝜅^𝑛−1/2𝜗^𝑛−1/2+𝛾𝜋^𝑛−1/2𝜅^𝑛−1/2+𝑓𝑢^𝑛−1/2−𝑓𝑢_ℎ^𝑛−1/2+𝜋^𝑛−1/2𝜗^𝑛−1/2𝜏≤𝐶^𝑛−𝜏^𝑛−1Δ𝑡²+𝑅₁^𝑛−1/22+𝑅₂^𝑛−1/22+𝜋^𝑛−1/22+𝛾2𝜅^𝑛−1/22+𝐶𝑓𝑢^𝑛−1/2−𝑓𝑢_ℎ^𝑛−1/22+𝐶𝜗^𝑛−1/22. (54)
In order to accomplish our process, we have to estimate the nonlinear term ‖𝑓(𝑢^𝑛−1/2)−𝑓(𝑢_ℎ^𝑛−1/2)‖². Using Young inequality, we have 𝑓𝑢^𝑛−1/2−𝑓𝑢_ℎ^𝑛−1/22≤2𝑢^𝑛−1/23−𝑢_ℎ^𝑛−1/232+2𝑢^𝑛−1/2−𝑢_ℎ^𝑛−1/223=22𝑢^𝑛−1−𝑢_ℎ^𝑛−1−12𝑢^𝑛−2−𝑢_ℎ^𝑛−2×32𝑢^𝑛−1−12𝑢^𝑛−22+32𝑢_ℎ^𝑛−1−12𝑢_ℎ^𝑛−22+32𝑢^𝑛−1−12𝑢^𝑛−232𝑢_ℎ^𝑛−1−12𝑢_ℎ^𝑛−223+22𝑢^𝑛−1−𝑢_ℎ^𝑛−1−12𝑢^𝑛−2−𝑢_ℎ^𝑛−223≤22𝜗^𝑛−1+𝜏^𝑛−1−12𝜗^𝑛−2+𝜏^𝑛−22×32𝑢^𝑛−1−12𝑢^𝑛−22+32𝑢_ℎ^𝑛−1−12𝑢_ℎ^𝑛−22+32𝑢^𝑛−1−12𝑢^𝑛−232𝑢_ℎ^𝑛−1−12𝑢_ℎ^𝑛−22_𝐿∞=+124𝐶²_𝑢3+22𝜗^𝑛−1+𝜏^𝑛−1−12𝜗^𝑛−2+𝜏^𝑛−22≤𝐶𝜏^𝑛−12+𝜏^𝑛−22+𝜗^𝑛−12+𝜗^𝑛−22. (55)
Note that 𝜏^𝑛−𝜏^𝑛−1Δ𝑡²≤1Δ𝑡^𝑡𝑛_𝑡𝑛−1𝜏_𝑡(𝑠)²𝑑𝑠. (56)
Substitute (55) and (56) into (54) to obtain 𝛾2𝜅^𝑛−1/22+1𝜗2Δ𝑡^𝑛2−𝜗^𝑛−121≤𝐶Δ𝑡^𝑡𝑛_𝑡𝑛−1𝜏_𝑡(𝑠)²𝑑𝑠+𝑅₁^𝑛−1/22+𝑅₂^𝑛−1/22+𝜋^𝑛−1/22+𝐶𝜏^𝑛−12+𝜏^𝑛−22+𝜗^𝑛2+𝜗^𝑛−12+𝜗^𝑛−22. (57)
Summing from 𝑛=2,…,𝐽, the resulting inequality becomes 𝛾Δ𝑡^𝐽_𝑛=2𝜅^𝑛−1/22+(1−𝐶Δ𝑡)𝜗^𝐽2≤𝜗¹²+𝐶Δ𝑡^𝐽_𝑛=2𝑅₁^𝑛−1/22+𝑅₂^𝑛−1/22+𝜋^𝑛−1/22+𝐶Δ𝑡^𝐽−1_𝑛=0𝜏^𝑛2+𝜗^𝑛2+𝐶^𝑡𝐽₀𝜏_𝑡²𝑑𝑠. (58)
For sufficiently small Δ𝑡, using Gronwall lemma with Lemma 10, we get 𝛾Δ𝑡^𝐽_𝑛=2𝜅^𝑛−1/22+12𝜗^𝐽2≤𝜗¹²+𝐶Δ𝑡^𝐽−1_𝑛=0𝜏^𝑛2+𝐶Δ𝑡^𝐽_𝑛=2𝑅₁^𝑛−1/22+𝑅₂^𝑛−1/22+𝜋^𝑛−1/22+𝐶^𝑡𝐽₀𝜏_𝑡²𝑑𝑠≤𝜗¹²+𝐶𝜏^𝑛2_𝐿∞𝐿2+𝜋^𝑛−1/22_𝐿∞𝐿2+^𝑡𝐽₀𝜏_𝑡²𝑑𝑠+𝐶Δ𝑡⁴𝑢_𝑡𝑡𝑡²_𝐿∞𝐿2. (59)
Use Lemmas 3 and 4, (59) with the triangle inequality to accomplish the proof of Theorem 12.

Theorem 13. With 𝑢⁰_ℎ=𝒬_ℎ𝑢⁰, assume that the solution's regular properties for system (6a), (6b), and (6c) satisfy 𝑢_𝑡∈𝐿²(𝐻²), 𝑢_𝑡𝑡𝑡∈𝐿^∞(𝐿²), 𝑢,𝑤∈𝐿^∞(𝐻²), 𝝌∈𝐿^∞(𝐇¹), 𝝌_𝑡∈𝐿²(𝐇¹). Then there exists a positive constant 𝐶 independent of ℎ and Δ𝑡 such that 𝑢𝑡_𝐽−𝑢^𝐽_ℎ1≤𝐶ℎ‖|⋅|‖+Δ𝑡²𝑢_{𝑡𝑡𝑡𝐿∞(𝐿2)},Δ𝑡^𝐽_𝑛=1𝑤𝑡_𝑛−1/2−𝑤_ℎ^𝑛−1/22_𝑗^1/2≤𝐶ℎ^2−𝑗‖|⋅|‖+Δ𝑡²𝑢_{𝑡𝑡𝑡𝐿∞(𝐿2)}𝝌𝑡,𝑗=0,1,_𝑛−1/2−𝝌_ℎ^𝑛−1/2≤𝐶ℎ‖|⋅|‖+Δ𝑡²𝑢_{𝑡𝑡𝑡𝐿∞(𝐿2)}. (60a)(60b)(60c)

Proof. Choose 𝑣_ℎ=𝜅^𝑛−1/2, 𝑔_ℎ=(𝜗^𝑛−𝜗^𝑛−1)/Δ𝑡, and 𝐳_ℎ=(∇𝜗^𝑛−∇𝜗^𝑛−1)/Δ𝑡 in (50a), (50b), and (50c) to obtain 𝜗^𝑛−𝜗^𝑛−1Δ𝑡,𝜅^𝑛−1/2+𝛾∇𝜅^𝑛−1/22+𝜅^𝑛−1/22+𝑓𝑢^𝑛−1/2−𝑓𝑢_ℎ^𝑛−1/2,𝜅^𝑛−1/2𝜏=−^𝑛−𝜏^𝑛−1Δ𝑡,𝜅^𝑛−1/2−𝜋^𝑛−1/2,𝜅^𝑛−1/2+𝑅₁^𝑛−1/2,𝜅^𝑛−1/2+𝑅₂^𝑛−1/2,𝜅^𝑛−1/2,𝜻^𝑛−1/2,∇𝜗^𝑛−∇𝜗^𝑛−1−𝜅Δ𝑡^𝑛−1/2,𝜗^𝑛−𝜗^𝑛−1=𝜋Δ𝑡_𝑛−1/2,𝜗^𝑛−𝜗^𝑛−1,−𝜻Δ𝑡_𝑛−1/2,∇𝜗^𝑛−∇𝜗^𝑛−1+Δ𝑡∇𝜗^𝑛2−∇𝜗^𝑛−122Δ𝑡=0. (61a)(61b)(61c)
We add the above three equations of (61a), (61b), and (61c) to get ∇𝜗^𝑛2−∇𝜗^𝑛−122Δ𝑡+𝛾∇𝜅^𝑛−1/22+𝜅^𝑛−1/22=−𝑓𝑢^𝑛−1/2−𝑓𝑢_ℎ^𝑛−1/2,𝜅^𝑛−1/2−𝜏^𝑛−𝜏^𝑛−1Δ𝑡,𝜅^𝑛−1/2−𝜋^𝑛−1/2,𝜅^𝑛−1/2+𝑅₁^𝑛−1/2,𝜅^𝑛−1/2+𝑅₂^𝑛−1/2,𝜅^𝑛−1/2+𝜋^𝑛−1/2,𝜗^𝑛−𝜗^𝑛−1𝑓𝑢Δ𝑡=−^𝑛−1/2−𝑓𝑢_ℎ^𝑛−1/2,𝜅^𝑛−1/2−𝜏^𝑛−𝜏^𝑛−1Δ𝑡,𝜅^𝑛−1/2−𝜋^𝑛−1/2,𝜅^𝑛−1/2+𝑅₁^𝑛−1/2,𝜅^𝑛−1/2+𝑅₂^𝑛−1/2,𝜅^𝑛−1/2+𝜋^𝑛,𝜗^𝑛−𝜋^𝑛−1,𝜗^𝑛−1−𝜗Δ𝑡^𝑛−1/2,𝜋^𝑛−𝜋^𝑛−1.Δ𝑡 (62)
For (62), we employ Cauchy-Schwarz inequality as well as Young inequality to obtain ∇𝜗^𝑛2−∇𝜗^𝑛−122Δ𝑡+𝛾∇𝜅^𝑛−1/22+12𝜅^𝑛−1/22≤𝐶𝑓𝑢^𝑛−1/2−𝑓𝑢_ℎ^𝑛−1/22+𝜏^𝑛−𝜏^𝑛−1Δ𝑡²+𝜋^𝑛−1/22+𝑅₁^𝑛−1/22+𝑅₂^𝑛−1/22+𝜗^𝑛−1/22+𝜋^𝑛−𝜋^𝑛−1Δ𝑡²+𝜋^𝑛,𝜗^𝑛−𝜋^𝑛−1,𝜗^𝑛−1.Δ𝑡 (63)
Multiply (63) by 2Δ𝑡 and then sum from 𝑛=2,…,𝐽 to get ∇𝜗^𝐽2+2𝛾Δ𝑡^𝐽_𝑛=2∇𝜅^𝑛−1/22+Δ𝑡^𝐽_𝑛=2𝜅^𝑛−1/22≤∇𝜗¹²+𝐶Δ𝑡^𝐽_𝑛=2𝑓𝑢^𝑛−1/2−𝑓𝑢_ℎ^𝑛−1/22+𝜏^𝑛−𝜏^𝑛−1Δ𝑡²+𝜋^𝑛−1/22+𝑅₁^𝑛−1/22+𝑅₂^𝑛−1/22+𝜗^𝑛−1/22+𝜋^𝑛−𝜋^𝑛−1Δ𝑡²+𝜋^𝐽,𝜗^𝐽−𝜋¹,𝜗¹. (64)
Substituting the following inequality 𝜋^𝑛−𝜋^𝑛−1Δ𝑡²≤1Δ𝑡^𝑡𝑛_𝑡𝑛−1𝜋_𝑡(𝑠)²𝑑𝑠, (65)
and (55) into (64), we obtain ∇𝜗^𝐽2+2𝛾Δ𝑡^𝐽_𝑛=2∇𝜅^𝑛−1/22+Δ𝑡^𝐽_𝑛=2𝜅^𝑛−1/22≤𝜗¹²₁+𝜋^𝐽2+𝐶Δ𝑡^𝐽_𝑛=2𝜏^𝑛−12+𝜏^𝑛−22+𝜗^𝑛−12+𝜗^𝑛−22+𝜋^𝑛−1/22+𝑅₁^𝑛−1/22+𝑅₂^𝑛−1/22+𝜗^𝑛−1/22+𝜗^𝐽2. (66)
Substitute (59) into (66) to obtain ∇𝜗^𝐽2+2𝛾Δ𝑡^𝐽_𝑛=2∇𝜅^𝑛−1/22+Δ𝑡^𝐽_𝑛=2𝜅^𝑛−1/22≤𝜗¹²₁+𝐶Δ𝑡^𝐽_𝑛=2𝜏^𝑛−12+𝜏^𝑛−22+𝜋^𝑛−1/22+𝑅₁^𝑛−1/22+𝑅₂^𝑛−1/22+𝜋^𝐽2+𝐶𝜏^𝑛2_𝐿∞𝐿2+^𝑡𝐽₀𝜏_𝑡²𝑑𝑠+Δ𝑡⁴𝑢_{𝑡𝑡𝑡𝐿∞(𝐿2)}+𝜋^𝑛−1/22_𝐿∞𝐿2. (67)
Taking 𝐳_ℎ=𝜻^𝑛−1/2 in (50c), we have 𝜻^𝑛−1/22≤𝐶∇𝜗^𝑛−1/22≤𝐶𝜗^𝑛2+𝜗^𝑛−12. (68)
Combining Lemmas 3–10, (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 take 𝛾=0 in (1), we can get the following second-order nonlinear reaction-diffusion equation: 𝑢_𝑡−Δ𝑢+𝑓(𝑢)=0,(𝐱,𝑡)∈Ω×𝐽, (69)

with initial condition 𝑢(𝐱,0)=𝑢₀(𝐱),𝐱∈Ω, (70)

and Dirichlet boundary conditions 𝑢(𝐱,𝑡)=0,(𝐱,𝑡)∈𝜕Ω×𝐽, (71)

where Ω is a bounded convex polygonal domain in 𝑅^𝑑, 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): 𝑢_𝑡,𝑣+(𝝀,∇𝑣)+𝑓(𝑢),𝑣=0,∀𝑣∈𝐻¹₀,(𝝀,𝐳)−(∇𝑢,𝐳)=0,∀𝐳∈𝐋², (72a)(72b)

where 𝝀=∇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].

Acknowledgments

This work was supported by the National Natural Science Fund of China (11061021), the Key Project of Chinese Ministry of Education (12024), the Scientific Research Projection of Higher Schools of Inner Mongolia (NJZZ12011, NJZY13199) and the Natural Science Fund of Inner Mongolia Province (2012MS0108, 2012MS0106, and 2011BS0102), the Program of Higher-level talents of Inner Mongolia University (125119, Z200901004, and 30105-125132), and the Scientific Research Projection of Inner Mongolia University of Finance and Economics (KY1101).