Crank-Nicolson Fully Discrete 𝐻¹-Galerkin Mixed Finite Element Approximation of One Nonlinear Integrodifferential Model

Abstract

We consider a fully discrete 𝐻¹-Galerkin mixed finite element approximation of one nonlinear integrodifferential model which often arises in mathematical modeling of the process of a magnetic field penetrating into a substance. We adopt the Crank-Nicolson discretization for time derivative. Optimal order a priori error estimates for the unknown function in 𝐿² and 𝐻¹ norm and its gradient function in 𝐿² norm are presented. A numerical example is given to verify the theoretical results.

1. Introduction

The objective of this paper is to discuss a Crank-Nicolson fully discrete 𝐻¹-mixed finite element scheme for the following nonlinear integrodifferential model: 𝑢_𝑡−(1+𝜆(𝑡))𝑢_𝑥𝑥(),()(],𝑢()()𝑢()=𝑓𝑥,𝑡𝑥,𝑡∈𝐼×0,𝑇0,𝑡=0,𝑢1,𝑡=0,0≤𝑡≤𝑇,𝑥,0=𝑢₀(𝑥),𝑥∈𝐼, (1) where 𝜆(𝑡)=∫^𝑡₀∫¹₀(𝜕𝑢/𝜕𝑥)²𝑑𝑥𝑑𝑠 and 𝐼=[0,1]. 𝑢₀(𝑥) and 𝑓(𝑥,𝑡) are given functions.

The above equations have been widely used to describe the process of a magnetic field penetrating into a substance, which is a generalization of the model proposed in [1–4]. The existence and uniqueness of a weak solution to the above boundary value problems were proved in [5].

During the last decades, many numerical methods were developed to discretize this kind of problems. For the finite difference approximation of the above model one can refer to [6–11]. For Galerkin finite element approximation of model (1) we can refer to [11], where the authors developed error estimates for semidiscretization in the energy norm. Note that the coefficient in (1) depends on the derivative of 𝑢. When finite difference method and Galerkin method were used to solve this model, one needed to differentiate the numerical solution to determine the coefficient. This would generate an inaccurate coefficient, which then reduces the accuracy of the numerical approximation for 𝑢. In order to overcome this question an 𝐻¹-Galerkin mixed finite element discrete scheme was proposed in [12]. Optimal order error estimates in 𝐿² norm and 𝐻¹ norm were presented. For more references with respect to 𝐻¹-Galerkin mixed finite element method one can refer to [13–17].

In [12] the backward Euler method was used to discretize the time derivative. Note that problem (1) is nonlocal due to the integration term in the coefficient. To improve the convergence order for time discretization and save the storage we construct a Crank-Nicolson 𝐻¹-mixed finite element scheme for problem (1). By using elliptic projection and the boundness of the numerical solutions we prove optimal a priori error estimates for the scalar unknown function and its flux. Finally we carry out a numerical example to verify our theoretical results.

The rest of this paper is organized as follows. In Section 2 a Crank-Nicolson 𝐻¹-mixed finite element scheme is constructed. Optimal a priori error estimates are deduced in Section 3. In Section 4 a numerical example is carried out to verify our theoretical results.

Throughout the paper, we use the standard notation 𝑊^𝑚,𝑞(Ω) for Sobolev space on Ω with a norm ‖⋅‖_𝑚,𝑞 and a seminorm |⋅|_𝑚,𝑞. For 𝑞=2, we denote 𝐻^𝑚(Ω)=𝑊^𝑚,2(Ω), ‖⋅‖_𝑚=‖⋅‖_𝑚,2, and for 𝑚=0, we denote ‖⋅‖=‖⋅‖₀. Moreover, the inner products in 𝐿²(Ω) are indicated by (⋅,⋅). Let 𝑋 be a Banach space and 𝜑(𝑡)∶[0,𝑇]↦𝑋; we set 𝜑²_𝐿2(𝑋)=^𝑇₀𝜑(𝑠)²_𝑋𝑑𝑠,𝜑_{𝐿∞(𝑋)}=esssup_{0≤𝑡≤𝑇}𝜑_𝑋. (2)

In addition, 𝐶 denotes a generic constant independent of the spatial mesh parameter ℎ and time discretization parameter 𝜏, and 𝜀 denotes an arbitrarily small positive constant.

2. Crank-Nicolson Discrete Scheme

In this section we first briefly describe the weak formulation for problem (1) and then construct a Crank-Nicolson discrete scheme for it.

2.1. Weak Formulation

In order to define a fully discrete 𝐻¹-Galerkin mixed finite element procedure for problem (1), we firstly split (1) into a first order system. Let 𝜎=𝑢_𝑥; then (1) reduces to 𝜎=𝑢_𝑥,𝑢_𝑡=(1+𝜆(𝑡))𝜎_𝑥+𝑓(𝑥,𝑡), (3) where 𝜆(𝑡)=∫^𝑡₀∫¹₀𝜎²𝑑𝑥𝑑𝑠.

Let 𝐻¹₀(𝐼)={𝑣∈𝐻¹(𝐼);𝑣(0)=𝑣(1)=0}. It is natural to state the weak formulation for problem (1) in the following form: 𝑢_𝑥,𝑣_𝑥=𝜎,𝑣_𝑥,𝑣∈𝐻¹₀(𝐼),𝜎_𝑡,𝑤+(1+𝜆(𝑡))𝜎_𝑥,𝑤_𝑥+𝑓,𝑤_𝑥=0,𝑤∈𝐻¹(𝐼). (4)

2.2. The Crank-Nicolson Discrete Scheme

First we introduce two finite element spaces. Let 𝑉_ℎ and 𝑊_ℎ denote the finite dimensional subspaces of 𝐻¹₀(𝐼) and 𝐻¹(𝐼), respectively, with the following approximation properties: inf_{𝜓ℎ∈𝑉ℎ}𝜓−𝜓_ℎ0,𝑝+ℎ𝜓−𝜓_ℎ1,𝑝≤𝐶ℎ^𝑘+1𝜓_𝑘+1,𝑝,𝜓∈𝐻¹₀(𝐼)∩𝑊^𝑘+1,𝑝(𝐼),inf_{𝑤ℎ∈𝑊ℎ}𝑤−𝑤_ℎ0,𝑝+ℎ𝑤−𝑤_ℎ1,𝑝≤𝐶ℎ^𝑟+1‖𝑤‖_𝑟+1,𝑝,𝑤∈𝑊^𝑟+1,𝑝(𝐼), (5) where 1≤𝑝≤∞. 𝑘, 𝑟 are positive integers.

To define the fully discrete scheme we also need a time mesh grid. Let 0=𝑡⁰<𝑡¹<⋯<𝑡^𝑁=𝑇 be a given partition of the time interval [0,𝑇] with step length 𝜏=𝑇/𝑁, for some positive integers 𝑁. Define 𝑡^𝑛=𝑛𝜏 and 𝑡^𝑛−(1/2)=(𝑛−(1/2))𝜏. For convenience we set 𝜙^𝑛=𝜙(𝑡^𝑛) and 𝜙^𝑛−(1/2)=(𝜙^𝑛+𝜙^𝑛−1)/2 for a smooth function 𝜙.

Let 𝑈^𝑛 and 𝑄^𝑛 denote the discrete counterpart of 𝑢 and 𝜎 at 𝑡=𝑡^𝑛 which satisfy the following Crank-Nicolson discrete scheme: 𝑈^𝑛_𝑥,𝑣_ℎ𝑥=𝑄^𝑛,𝑣_ℎ𝑥,𝑣_ℎ∈𝑉_ℎ,𝑄^𝑛−𝑄^𝑛−1𝜏,𝑤_ℎ+1+^𝑛_𝑖=1𝛿_𝑛,𝑖𝑄^𝑖2+𝑄^𝑖−122𝑄^𝑛_𝑥+𝑄_𝑥^𝑛−12,𝑤_ℎ𝑥+𝑓^𝑛−(1/2),𝑤_ℎ𝑥=0,𝑤_ℎ∈𝑊_ℎ, (6)(7) where 𝛿_𝑛,𝑖=𝜏𝜏,𝑖=1,2,…,𝑛−1,2,𝑖=𝑛, (8) and 𝑈⁰, 𝑄⁰ are to be defined later.

The existence and uniqueness of the discrete solution for the above problems can be guaranteed by the theory presented in [18, page 237–239].

To discretize the time integration we used the following integroformula: ^{𝑡𝑛−(1/2)}₀𝑔(𝜏)𝑑𝑠≈^𝑛_𝑖=1𝛿_𝑛,𝑖𝑔^𝑖−(1/2). (9) Its truncation error can be estimated as follows: ^{𝑡𝑛−(1/2)}₀𝑔(𝜏)𝑑𝑠−^𝑛_𝑖=1𝛿_𝑛,𝑖𝑔^𝑛−(1/2)≤𝐶𝜏^𝑡𝑛_𝑡𝑛−1𝑔_𝑡+𝜏^2𝑡𝑛₀𝑔_𝑡𝑡𝑑𝑠. (10)

3. Error Analysis

3.1. Preliminaries

We begin by recalling some preliminary knowledge that will be used in the following convergence analysis.

We define the following elliptic projections: 𝑢_ℎ(𝑡)∈𝑉_ℎ, 𝜎_ℎ(𝑡)∈𝑊_ℎ, which satisfy 𝑢_𝑥−𝑢_ℎ𝑥,𝑣_ℎ𝑥=0,∀𝑣_ℎ∈𝑉_ℎ,𝜎_𝑥−𝜎_ℎ𝑥,𝑤_ℎ𝑥+𝛼𝜎−𝜎_ℎ,𝑤_ℎ=0,∀𝑤_ℎ∈𝑊_ℎ. (11) Here 𝛼 is chosen to guarantee the 𝐻¹-coercivity of the bilinear form in the second equations. Moreover, it is easy to check that the bilinear form is bounded.

Let 𝜂=𝑢−𝑢_ℎ, 𝜌=𝜎−𝜎_ℎ; then 𝜂 and 𝜌 satisfy the following estimates from [19]: 𝜂(𝑡)_𝑗+𝜂_𝑡(𝑡)_𝑗≤𝐶ℎ^{𝑘+1−𝑗}‖𝑢‖_𝑘+1+𝑢_𝑡𝑘+1𝜌(𝑡,𝑗=0,1,)_𝑗+𝜌_𝑡(𝑡)_𝑗≤𝐶ℎ^{𝑟+1−𝑗}𝑞_𝑟+1+𝑞_𝑡𝑟+1,𝑗=0,1. (12)

To derive the error estimates we also need the following discrete Gronwall inequality.

Lemma 1 (discrete Gronwall inequality; see [20]). Let 𝜏,𝐵₁,𝐶₁>0 and let 𝑎_𝑛, 𝑏_𝑛, 𝑐_𝑛, and 𝑑_𝑛 be sequences of nonnegative numbers satisfying ∀𝑛≥0,𝑎_𝑛+𝜏^𝑛_𝑖=0𝑏_𝑖≤𝐵₁+𝐶₁𝜏^𝑛_𝑖=0𝑎_𝑖+𝜏^𝑛_𝑖=0𝑐_𝑖. (13) Then, if 𝐶₁𝜏<1, ∀𝑛≥0,𝑎_𝑛+𝜏^𝑛_𝑖=0𝑏_𝑖≤𝑒^{𝐶1(𝑛+1)𝜏}𝐵₁+𝜏^𝑛_𝑖=0𝑐_𝑖. (14)

3.2. Error Analysis

To estimate the errors, we firstly decompose the errors into 𝑢𝑡^𝑛−𝑈^𝑛=𝑢𝑡^𝑛−𝑢_ℎ𝑡^𝑛+𝑢_ℎ𝑡^𝑛−𝑈^𝑛=𝜂^𝑛+𝜁^𝑛,𝜎𝑡^𝑛−𝑄^𝑛=𝜎𝑡^𝑛−𝜎_ℎ𝑡^𝑛+𝜎_ℎ𝑡^𝑛−𝑄^𝑛=𝜌^𝑛+𝜉^𝑛. (15) Note that the estimates of 𝜂^𝑛 and 𝜌^𝑛 can be found out easily from (12) at 𝑡=𝑡^𝑛. Therefore it remains to estimate 𝜁^𝑛 and 𝜉^𝑛.

Setting 𝑡=𝑡^𝑛−(1/2) in (4) and combining (6) and (7) with auxiliary projections, we deduce the following error equations with respect to 𝜁^𝑛 and 𝜉^𝑛: 𝜁^𝑛_𝑥,𝑣_ℎ𝑥=𝜌^𝑛,𝑣_ℎ𝑥+𝜉^𝑛,𝑣_ℎ𝑥,𝜉^𝑛−𝜉^𝑛−1𝜏,𝑤_ℎ+𝜉^𝑛_𝑥+𝜉_𝑥^𝑛−12,𝑤_ℎ𝑥=𝜎^𝑛−𝜎^𝑛−1𝜏−𝜎_𝑡^𝑛−(1/2),𝑤_ℎ−𝜌^𝑛−𝜌^𝑛−1𝜏,𝑤_ℎ𝜌+𝛼^𝑛+𝜌^𝑛−12,𝑤_ℎ−𝜆^𝑛−(1/2)𝜎_𝑥^𝑛−(1/2)−^𝑛_𝑖=1𝛿_𝑛,𝑖𝑄^𝑖2+𝑄^𝑖−122×𝑄^𝑛_𝑥+𝑄_𝑥^𝑛−12,𝑤_ℎ𝑥+𝜎^𝑛_𝑥+𝜎_𝑥^𝑛−12−𝜎_𝑥^𝑛−(1/2),𝑤_ℎ𝑥. (16)(17)

Theorem 2. Suppose that 𝑈⁰=𝑢_ℎ(0), 𝑄⁰=𝜎_ℎ(0), and 1≤𝐽≤𝑁. Then there exists a positive constant C independent of h and 𝜏 such that for sufficiently small 𝜏𝑢^𝐽−𝑈^𝐽+ℎ𝑢^𝐽−𝑈^𝐽₁+𝜎^𝐽−𝑄^𝐽≤𝐶ℎ^{min{𝑘+1,𝑟+1}}+𝜏². (18) Here 𝑘,𝑟≥1 are positive integers.

Proof. Choosing 𝑣_ℎ=𝜁^𝑛 in (16) yields 𝜁^𝑛_𝑥,𝜁^𝑛_𝑥=𝜌^𝑛,𝜁^𝑛_𝑥+𝜉^𝑛,𝜁^𝑛_𝑥, (19) which implies 𝜁^𝑛_𝑥≤𝜌^𝑛+𝜉^𝑛. (20) Setting 𝑤_ℎ=(𝜉^𝑛+𝜉^𝑛−1)/2 in (17) gives 𝜉^𝑛−𝜉^𝑛−1𝜏,𝜉^𝑛+𝜉^𝑛−12+𝜉^𝑛_𝑥+𝜉_𝑥^𝑛−12,𝜉^𝑛_𝑥+𝜉_𝑥^𝑛−12+𝜆^𝑛−(1/2)𝜎_𝑥^𝑛−(1/2)−^𝑛_𝑖=1𝛿_𝑛,𝑖𝑄^𝑖2+𝑄^𝑖−122×𝑄^𝑛_𝑥+𝑄_𝑥^𝑛−12,𝜉^𝑛_𝑥+𝜉_𝑥^𝑛−12=𝜎^𝑛−𝜎^𝑛−1𝜏−𝜎_𝑡^𝑛−(1/2),𝜉^𝑛+𝜉^𝑛−12−𝜌^𝑛−𝜌^𝑛−1𝜏,𝜉^𝑛+𝜉^𝑛−12𝜌+𝛼^𝑛+𝜌^𝑛−12,𝜉^𝑛+𝜉^𝑛−12+𝜎^𝑛_𝑥+𝜎_𝑥^𝑛−12−𝜎_𝑥^𝑛−(1/2),𝜉^𝑛_𝑥+𝜉_𝑥^𝑛−12. (21)
For the third term on the left side we have 𝜆^𝑛−(1/2)𝜎_𝑥^𝑛−(1/2)−^𝑛_𝑖=1𝛿_𝑛,𝑖𝑄^𝑖2+𝑄^𝑖−122×𝑄^𝑛_𝑥+𝑄_𝑥^𝑛−12,𝜉^𝑛_𝑥+𝜉_𝑥^𝑛−12=^𝑛_𝑖=1𝛿_𝑛,𝑖𝑄^𝑖2+𝑄^𝑖−122𝜉^𝑛_𝑥+𝜉_𝑥^𝑛−12,𝜉^𝑛_𝑥+𝜉_𝑥^𝑛−12+𝜆^𝑛−(1/2)−^𝑛_𝑖=1𝛿_𝑛,𝑖𝑄^𝑖2+𝑄^𝑖−122𝜎_𝑥^𝑛−(1/2),𝜉^𝑛_𝑥+𝜉_𝑥^𝑛−12+^𝑛_𝑖=1𝛿_𝑛,𝑖𝑄^𝑖2+𝑄^𝑖−122𝜎_𝑥^𝑛−(1/2)−𝜎^𝑛_𝑥+𝜎_𝑥^𝑛−12,𝜉^𝑛_𝑥+𝜉_𝑥^𝑛−12−𝛼^𝑛_𝑖=1𝛿_𝑛,𝑖𝑄^𝑖2+𝑄^𝑖−122𝜌^𝑛+𝜌^𝑛−12,𝜉^𝑛+𝜉^𝑛−12. (22) Then we obtain 1𝜉2𝜏^𝑛2−𝜉^𝑛−12+1+^𝑛_𝑖=1𝛿_𝑛,𝑖𝑄^𝑖2+𝑄^𝑖−122×𝜉^𝑛_𝑥+𝜉_𝑥^𝑛−12²=−𝜆^𝑛−(1/2)−^𝑛_𝑖=1𝛿_𝑛,𝑖𝑄^𝑖2+𝑄^𝑖−122𝜎_𝑥^𝑛−(1/2),𝜉^𝑛_𝑥+𝜉_𝑥^𝑛−12+^𝑛_𝑖=1𝛿_𝑛,𝑖𝑄^𝑖2+𝑄^𝑖−122𝜎^𝑛_𝑥+𝜎_𝑥^𝑛−12−𝜎_𝑥^𝑛−(1/2),𝜉^𝑛_𝑥+𝜉_𝑥^𝑛−12+𝜎^𝑛−𝜎^𝑛−1𝜏−𝜎_𝑡^𝑛−(1/2),𝜉^𝑛+𝜉^𝑛−12−𝜌^𝑛−𝜌^𝑛−1𝜏,𝜉^𝑛+𝜉^𝑛−12+𝛼1+^𝑛_𝑖=1𝛿_𝑛,𝑖𝑄^𝑖2+𝑄^𝑖−122𝜌^𝑛+𝜌^𝑛−12,𝜉^𝑛+𝜉^𝑛−12+𝜎^𝑛_𝑥+𝜎_𝑥^𝑛−12−𝜎^𝑛−(1/2)_ℎ𝑥,𝜉^𝑛_𝑥+𝜉_𝑥^𝑛−12. (23) Now it remains to estimate the terms on the right side of (23). It is easy to prove that 𝜌^𝑛−𝜌^𝑛−1𝜏²1≤𝐶𝜏^𝑡𝑛_𝑡^𝑛−1𝜌_𝑡²𝜎𝑑𝑠,_𝑛−𝜎^𝑛−1𝜏−𝜎_𝑡^{𝑛−(1/2)2}≤𝐶𝜏^3𝑡𝑛_𝑡𝑛−1𝜎_𝑡𝑡𝑡²𝜎𝑑𝑠,_𝑛𝑥+𝜎_𝑥^𝑛−12−𝜎_𝑥^{𝑛−(1/2)2}≤𝐶𝜏^3𝑡𝑛_𝑡𝑛−1𝜎_{𝑥𝑡𝑡𝑡}²𝑑𝑠. (24) Here the Taylor formula with integral remainder was used.
Note that 𝜆^𝑛−(1/2)−^𝑛_𝑖=1𝛿_𝑛,𝑖𝑄^𝑖2+𝑄^𝑖−122=¹₀^{𝑡𝑛−(1/2)}₀𝜎²𝑑𝑠−^𝑛_𝑖=1𝛿_𝑛,𝑖𝑄^𝑖2+𝑄^𝑖−122≤𝑑𝑥_{10𝑡𝑛−(1/2)0}𝜎²−𝜎²_ℎ+𝑑𝑠𝑑𝑥_{10𝑡𝑛−(1/2)0}𝜎²_ℎ𝑑𝑠−^𝑛_𝑖=1𝛿_𝑛,𝑖𝜎^𝑖_ℎ²+𝜎_ℎ^𝑖−122+𝑑𝑥_10𝑛𝑖=1𝛿_𝑛,𝑖𝜎^𝑖_ℎ²+𝜎_ℎ^𝑖−122−𝑄^𝑖2+𝑄^𝑖−122𝑑𝑥∶=𝐼₁+𝐼₂+𝐼₃. (25) By Hölder inequality and the bound of the projection 𝜎_ℎ we have 𝐼₁=¹₀^{𝑡𝑛−(1/2)}₀𝜎²−𝜎²_ℎ≤𝜎𝑑𝑠𝑑𝑥𝜎−_{ℎ𝐿2(0,𝑡𝑛;𝐿2(𝐼))}‖𝜎‖_𝐿²_(0,𝑡^𝑛_;𝐿²_(𝐼))+𝜎_{ℎ𝐿2(0,𝑡𝑛;𝐿2(𝐼))}≤𝐶𝜌_{𝐿2(0,𝑡𝑛;𝐿2(𝐼))}. (26) Using the error formula (10) and the bound of the projection 𝜎_ℎ we derive 𝐼₂=¹₀^{𝑡𝑛−(1/2)}₀𝜎²_ℎ𝑑𝑠−^𝑛_𝑖=1𝛿_𝑛,𝑖𝜎_ℎ^{𝑖−(1/2)2}𝑑𝑥≤𝐶¹₀𝜏^𝑡𝑛_𝑡𝑛−1𝑑𝜎𝑑𝑡^2ℎ𝑑𝑠+𝜏^2𝑡𝑛₀𝑑²𝑑𝑡²𝜎²_ℎ𝑑𝑠𝑑𝑥=𝐶¹₀𝜏^𝑡𝑛_𝑡𝑛−12𝜎_ℎ𝜎_ℎ𝑡𝑑𝑠+𝜏^2𝑡𝑛₀2𝜎²_ℎ𝑡+2𝜎_ℎ𝑡𝜎_ℎ𝑡𝑡𝑑𝑠𝑑𝑥≤𝐶𝜏^𝑡𝑛_𝑡𝑛−1𝜎_ℎ𝜎_ℎ𝑡𝑑𝑠+𝐶𝜏²𝜎_{ℎ𝑡𝐿2(0,𝑡𝑛;𝐿2(𝐼))}+𝜎_{ℎ𝐿2(0,𝑡𝑛;𝐿2(𝐼))}𝜎_{ℎ𝑡𝑡𝐿2(0,𝑡𝑛;𝐿2(𝐼))}≤𝐶𝜏^3/2𝑡𝑛_𝑡𝑛−1𝜎_ℎ²𝜎_ℎ𝑡²𝑑𝑠^1/2+𝜏². (27) For 𝐼₃ we have 𝐼₃=12¹₀^𝑛_𝑖=1𝛿_𝑛,𝑖𝜎^𝑖_ℎ²−𝑄^𝑖2+𝜎_ℎ^𝑖−12−𝑄^𝑖−12=1𝑑𝑥2¹₀^𝑛_𝑖=1𝛿_𝑛,𝑖𝜎^𝑖_ℎ−𝑄^𝑖𝜎^𝑖_ℎ+𝑄^𝑖+𝜎_ℎ^𝑖−1−𝑄^𝑖−1𝜎_ℎ^𝑖−1+𝑄^𝑖−1≤1𝑑𝑥2^𝑛_𝑖=1𝛿_𝑛,𝑖𝜉^𝑖𝜎^𝑖_ℎ+𝑄^𝑖+𝜉^𝑖−1𝜎_ℎ^𝑖−1+𝑄^𝑖−1. (28) To bound 𝐼₃ we need to derive the boundness of 𝑄^𝑖. From (7) we can deduce 1+^𝑛_𝑖=1𝛿_𝑛,𝑖𝑄^𝑖2+𝑄^𝑖−122𝑄^𝑛_𝑥+𝑄_𝑥^𝑛−12,𝑄^𝑛_𝑥+𝑄_𝑥^𝑛−12+𝑄^𝑛−𝑄^𝑛−1𝜏,𝑄^𝑛+𝑄^𝑛−12+𝑓^𝑛−(1/2),𝑄^𝑛_𝑥+𝑄_𝑥^𝑛−12=0. (29) By 𝜀 inequality we have 1𝑄2𝜏^𝑛2−𝑄^𝑛−12+12+^𝑛_𝑖=1𝛿_𝑛,𝑖𝑄^𝑖2+𝑄^𝑖−122×𝑄^𝑛_𝑥+𝑄_𝑥^𝑛−12²≤𝑓^{𝑛−(1/2)2}. (30) Then, multiplying by 2𝜏 and summing from 1 to 𝑙 we conclude 𝑄^𝑙2≤𝜏^𝑙_𝑖=1𝑓^{𝑛−(1/2)2}+𝑄⁰², (31) which implies 𝑄^𝑙 is bounded.
Combining (31) with the above estimate of 𝐼₃ and using the boundness of 𝜎_ℎ, we can get 𝐼₃≤𝐶𝜏^𝑛_𝑖=1𝜉^𝑖, (32) where 𝜉⁰=0 was used.
Collecting the above estimates for 𝐼₁∼𝐼₃ and using 𝜀 inequality, we obtain 𝜆^𝑛−(1/2)−^𝑛_𝑖=1𝛿_𝑛,𝑖𝑄^𝑖2+𝑄^𝑖−122𝜎_𝑥^𝑛−(1/2),𝜉^𝑛_𝑥+𝜉_𝑥^𝑛−12≤𝐶𝜌_{𝐿2(0,𝑡𝑛;𝐿2(𝐼))}+𝜏^3/2𝑡𝑛_𝑡𝑛−1𝜎_ℎ²𝜎_ℎ𝑡²𝑑𝑠^1/2+𝜏²×𝜉^𝑛_𝑥+𝜉_𝑥^𝑛−12+𝜏^𝑛_𝑖=1𝜉^𝑖𝜉^𝑛_𝑥+𝜉_𝑥^𝑛−12≤𝐶𝜌²_{𝐿20,𝑡𝑛;𝐿2(𝐼)}+𝜏^3𝑡𝑛_𝑡𝑛−1𝜎_ℎ²𝜎_ℎ𝑡²𝑑𝑠+𝜏⁴+𝜏^𝑛_𝑖=1𝜉^𝑖2𝜉+𝜀^𝑛𝑥+𝜉_𝑥^𝑛−12²≤𝐶𝜌²_{𝐿20,𝑡𝑛;𝐿2(𝐼)}+𝜏^3𝑡𝑛_𝑡𝑛−1𝜎_ℎ²𝜎_ℎ𝑡²𝑑𝑠+𝜏⁴+𝜏^𝑛_𝑖=1𝜉^𝑖2𝜉+𝜀^𝑛𝑥+𝜉_𝑥^𝑛−12²_𝑛𝑖=1,𝛿_𝑛,𝑖𝑄^𝑖2+𝑄^𝑖−122𝜎^𝑛_𝑥+𝜎_𝑥^𝑛−12−𝜎_𝑥^𝑛−(1/2),𝜉^𝑛_𝑥+𝜉_𝑥^𝑛−12𝜎≤𝐶^𝑛𝑥+𝜎_𝑥^𝑛−12−𝜎_𝑥^𝑛−(1/2)𝜉^𝑛_𝑥+𝜉_𝑥^𝑛−12𝜎≤𝐶^𝑛𝑥+𝜎_𝑥^𝑛−12−𝜎_𝑥^{𝑛−(1/2)2}𝜉+𝜀^𝑛𝑥+𝜉_𝑥^𝑛−12²≤𝐶𝜏^3𝑡𝑛_𝑡𝑛−1𝜎_{𝑥𝑡𝑡𝑡}²𝜉𝑑𝑠+𝜀^𝑛𝑥+𝜉_𝑥^𝑛−12². (33) Inserting (24) and (33) into (23) leads to 1𝜉2𝜏^𝑛2−𝜉^𝑛−12≤𝐶𝜌²_{𝐿20,𝑡𝑛;𝐿2(𝐼)}+𝜌^𝑛2+𝜌^𝑛−12+𝜏^3𝑡𝑛_𝑡𝑛−1𝜎_ℎ²𝑑𝑠+𝜏⁴+𝜏^𝑛_𝑖=1𝜉^𝑖21+𝐶𝜏^𝑡𝑛_𝑡^𝑛−1𝜌_𝑡²𝑑𝑠+𝐶𝜏^3𝑡𝑛_𝑡𝑛−1𝜎_{𝑥𝑡𝑡𝑡}²𝑑𝑠+𝐶𝜏^3𝑡𝑛_𝑡𝑛−1𝜎_𝑡𝑡𝑡²𝜉𝑑𝑠+𝐶^𝑛+𝜉^𝑛−12². (34) Multiplying by 2𝜏 and summing from 1 to 𝐽 leads to 𝜉^𝐽2≤𝐶𝜌²_{𝐿∞(0,𝑡𝐽;𝐿2(𝐼))}+𝐶𝜏^4𝑡𝐽₀𝜎_ℎ²𝑑𝑠+𝐶𝜏⁴+𝐶^𝑡𝐽₀𝜌_𝑡²𝑑𝑠+𝐶𝜏^4𝑡𝐽₀𝜎_{𝑥𝑡𝑡𝑡}²𝑑𝑠+𝐶𝜏^4𝑡𝐽₀𝜎_𝑡𝑡𝑡²𝑑𝑠+𝐶𝜏^2𝐽𝑛_𝑛=1𝑖=1𝜉^𝑖2+𝐶𝜏^𝐽_𝑛=1𝜉^𝑛+𝜉^𝑛−12²≤𝐶𝜌²_{𝐿∞(0,𝑡𝐽;𝐿2(𝐼))}+𝐶^𝑡𝐽₀𝜌_𝑡²𝑑𝑠+𝐶𝜏^𝐽_𝑛=1𝜉^𝑛2+𝐶𝜏^4𝑡𝐽₀𝜎_ℎ²𝑑𝑠+^𝑡𝐽₀𝜎_{𝑥𝑡𝑡𝑡}²+𝑑𝑠_𝑡𝐽0𝜎_𝑡𝑡𝑡²𝜌𝑑𝑠+1≤𝐶²_{𝐿∞(0,𝑡𝐽;𝐿2(𝐼))}+𝐶^𝑡𝐽₀𝜌_𝑡²𝑑𝑠+𝐶𝜏^𝐽_𝑛=1𝜉^𝑛2+𝐶𝜏⁴. (35) Taking 𝜏₁, let 0<𝜏≤𝜏₁, such that 1−𝐶𝜏>0; then by discrete Gronwall’s lemma we obtain that 𝜉^𝐽2≤𝐶𝜌²_{𝐿∞(0,𝑡𝐽;𝐿2(𝐼))}+𝐶^𝑡𝐽₀𝜌_𝑡²𝑑𝑠+𝐶𝜏⁴. (36) Substituting (36) into (20) yields 𝜁^𝐽_𝑥²≤𝐶𝜌²_{𝐿∞(0,𝑡𝐽;𝐿2(𝐼))}+𝐶^𝑡𝐽₀𝜌_𝑡²𝑑𝑠+𝐶𝜏⁴ (37) Combining (36), (37), and the estimates of 𝜂^𝑛, 𝜌^𝑛 and using the triangle inequality, we can complete the proof.

4. Numerical Example

In this section a numerical example is carried out to verify the theorems presented in this paper.

Example 1. Let us consider the initial and boundary problem (1) with the initial value 𝑢(𝑥,0)=𝑥(1−𝑥)sin(𝑥) and the right hand term ()()−𝑓=𝑥1−𝑥cos𝑥+𝑡1+11160𝑡−8(𝑡)(𝑡)−1cossin8()()+1cos𝑡+1sin𝑡+18(1)(1)×(()()()()()(.cossin−2sin𝑥+𝑡+21−𝑥cos𝑥+𝑡−2𝑥cos𝑥+𝑡−𝑥1−𝑥sin𝑥+𝑡)) (38) This example is taken from [11].

In this example we use piecewise linear finite element spaces to approximate the unknown functions 𝑢 and 𝜎, respectively. The Crank-Nicolson method is used to approximate the time derivative. Then the corresponding error estimates reduce to 𝑢^𝐽−𝑈^𝐽+ℎ𝑢^𝐽−𝑈^𝐽₁+𝜎^𝐽−𝑄^𝐽≤𝐶ℎ²+𝜏². (39) In the numerical implementation we choose ℎ=𝜏. The errors and the corresponding rate of convergence for 𝑢^𝐽−𝑈^𝐽 and 𝜎^𝐽−𝑄^𝐽 are displayed in Tables 1, 2, and 3, respectively.

Table 1 

The errors of ‖𝑢^𝐽−𝑈^𝐽‖ at different time.

Table 2 

The errors of ‖𝑢^𝐽_𝑥−𝑈^𝐽_𝑥‖ at different time.

Table 3 

The errors of ‖𝜎^𝐽−𝑄^𝐽‖ at different time.

We can observe that the numerical results are in agreement with our theoretical results proposed in Section 3.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.