Crank-Nicolson Fully Discrete H 1-Galerkin Mixed Finite Element Approximation of One Nonlinear Integrodifferential Model

and Applied Analysis 3 Let η = u−?̃? h ,ρ = σ−?̃? h ; then η andρ satisfy the following estimates from [19]: 󵄩󵄩󵄩󵄩η (t) 󵄩󵄩󵄩󵄩j + 󵄩󵄩󵄩󵄩ηt (t) 󵄩󵄩󵄩󵄩j ≤ Ch k+1−j (‖u‖k+1 + 󵄩󵄩󵄩󵄩ut 󵄩󵄩󵄩󵄩k+1 ) , j = 0, 1, 󵄩󵄩󵄩󵄩ρ (t) 󵄩󵄩󵄩󵄩j + 󵄩󵄩󵄩󵄩ρt (t) 󵄩󵄩󵄩󵄩j ≤ Ch r+1−j ( 󵄩󵄩󵄩󵄩q 󵄩󵄩󵄩󵄩r+1 + 󵄩󵄩󵄩󵄩qt 󵄩󵄩󵄩󵄩r+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, B 1 , C 1 > 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=0 b i ≤ B 1 + C 1 τ n ∑ i=0 a i + τ n ∑ i=0 c i . (13) Then, if C 1 τ < 1, ∀n ≥ 0, a n + τ n ∑ i=0 b i ≤ e C 1 (n+1)τ (B 1 + τ n ∑ i=0 c i ) . (14) 3.2. Error Analysis. To estimate the errors, we firstly decompose the errors into u (t n ) − U n = u (t n ) − ?̃? h (t n ) + ?̃? h (t n ) − U n = η n + ζ n , σ (t n ) − Q n = σ (t n ) − ?̃? h (t n ) + ?̃? h (t n ) − Q n = ρ n + ξ n . (15) Note that the estimates of η and ρ can be found out easily from (12) at t = t. Therefore it remains to estimate ζ and ξ. Setting t = t in (4) and combining (6) and (7) with auxiliary projections, we deduce the following error equations with respect to ζ and ξ: (ζ n x , V hx ) = (ρ n , V hx ) + (ξ n , V hx ) , (16)

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][2][3][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][7][8][9][10][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 . 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 . In order to overcome this question an 1 -Galerkin mixed finite element discrete scheme was proposed in [12]. Optimal order error estimates in 2 norm and 1 norm were presented. For more references with respect to 1 -Galerkin mixed finite element method one can refer to [13][14][15][16][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 1 -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 1 -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.

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.

Weak Formulation.
In order to define a fully discrete 1 -Galerkin mixed finite element procedure for problem (1), we firstly split (1) into a first order system. Let = ; then (1) reduces to = , where ( ) = ∫ 0 ∫ It is natural to state the weak formulation for problem (1) in the following form:

The Crank-Nicolson Discrete Scheme.
First we introduce two finite element spaces. Let ℎ and ℎ denote the finite dimensional subspaces of 1 0 ( ) and 1 ( ), respectively, with the following approximation properties: where 1 ≤ ≤ ∞. , are positive integers.
Let and denote the discrete counterpart of and at = which satisfy the following Crank-Nicolson discrete scheme: where , = and 0 , 0 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: Its truncation error can be estimated as follows:

Preliminaries.
We begin by recalling some preliminary knowledge that will be used in the following convergence analysis.
We define the following elliptic projections:̃ℎ( ) ∈ ℎ , ℎ ( ) ∈ ℎ , which satisfy Here is chosen to guarantee the 1 -coercivity of the bilinear form in the second equations. Moreover, it is easy to check that the bilinear form is bounded.
Let = −̃ℎ, = −̃ℎ; then and satisfy the following estimates from [19]: To derive the error estimates we also need the following discrete Gronwall inequality.

Error Analysis.
To estimate the errors, we firstly decompose the errors into Note that the estimates of and can be found out easily from (12) at = . Therefore it remains to estimate and . Setting = −(1/2) in (4) and combining (6) and (7) with auxiliary projections, we deduce the following error equations with respect to and : Theorem 2. Suppose that 0 =̃ℎ(0), 0 =̃ℎ(0), and 1 ≤ ≤ . Then there exists a positive constant C independent of h and such that for sufficiently small Here , ≥ 1 are positive integers.
Abstract and Applied Analysis 5 Using the error formula (10) and the bound of the projectioñ ℎ we derive +̃ℎ 2 (0, ; 2 ( ))̃ℎ 2 (0, ; 2 ( )) ) To bound 3 we need to derive the boundness of . From (7) we can deduce By inequality we have Then, multiplying by 2 and summing from 1 to we conclude which implies is bounded. Combining (31) with the above estimate of 3 and using the boundness of̃ℎ, we can get where 0 = 0 was used. Collecting the above estimates for 1 ∼ 3 and using inequality, we obtain Inserting (24) and (33) into (23) leads to Multiplying by 2 and summing from 1 to leads to Combining (36), (37), and the estimates of , and using the triangle inequality, we can complete the proof.

Numerical Example
In this section a numerical example is carried out to verify the theorems presented in this paper.
This example is taken from [11].
In this example we use piecewise linear finite element spaces to approximate the unknown functions and , respectively. The Crank-Nicolson method is used to approximate the time derivative. Then the corresponding error estimates reduce to In the numerical implementation we choose ℎ = . The errors and the corresponding rate of convergence for − and − are displayed in Tables 1, 2, and 3, respectively. We can observe that the numerical results are in agreement with our theoretical results proposed in Section 3.