1. Introduction

In this article, we consider the following nonlinear pseudo-hyperbolic equation [1-9]^[1] with initial and boundary conditions

(1)

where is a bounded domain in, which boundary is smooth. The initial values and are known functions, the coefficient is a bounded and smooth function. is a nonlinear bounded function of with and a bounded derivative.

As we know, the pseudo-hyperbolic equations, which describe many physical phenomena, such as heat and mass transfer and nerve conduction, are a kind of important hyperbolic wave equations. In view of the importance of the pseudo-hyperbolic equations, a lot of numerical methods have been studied, such as splitting positive definite mixed element method ^{[1, 2]}, least-squares mixed element method ^{[3, 4]}, H¹-Galerkin mixed element methods ^{[5, 6, 7, 8]}, mixed covolume method ^[9] and so forth.

Pani ^[10] presented and analyzed an H¹-Galerkin mixed method for solving the linear parabolic equation with initial and boundary condition and derived some semi-discrete and fully discrete a priori error results in detail. This method holds some good features: free of LBB condition, the freedom of approximation space’s selection and the better order of convergence for the flux in L²-norm. Based on the above value points, the H¹-Galerkin mixed element method for many evolution equations (pseudo-hyperbolic equations ^{[5, 6]}, integro-differential equation ^[11], hyperbolic equation ^[12], Sobolev equation ^[13], RLW equation ^{[14, 15]}, Schrodinger equation ^[16], Burgers’ equation ^[20]) were formulated. At the same time, some good numerical methods (Nonconforming H¹-Galerkin mixed method ^{[7, 17]}, Splitting H¹-Galerkin mixed method ^[8], H¹-Galerkin expanded mixed methods ^{[18, 19]}) based on the H¹-Galerkin mixed element method were developed by many authors. Liu and Li ^[5], Zhou ^[6] gave the theoretical analysis of a priori errors for H¹-Galerkin mixed method, respectively. Zhang et al. ^[7] studied a semi-discrete nonconforming H¹-Galerkin mixed scheme. Liu et al. ^[8] proposed a new splitting H¹-Galerkin mixed method, then analyzed a Crank-Nicolson fully discrete a priori error results. But in ^{[5, 6, 7, 8]}, the considered pseudo-hyperbolic equation is only a linear problem.

In this article, our aim is to consider a nonlinear pseudo-hyperbolic problem and to propose a fully discrete two-step difference method combining with an H¹-Galerkin mixed element method. Compared to backward Euler discrete method ^{[5, 6]} based on the linear problem, our method get some optimal second-order results of convergence in time direction for nonlinear pseudo-hyperbolic equation (1). As far as we know, the study of the H¹-Galerkin mixed method based on two-step difference scheme is fairly limited in the literatures, so our study is meaningful. We will give some a priori error analysis in detail and make some numerical simulation.

2. Two-Step Mixed Scheme and A Priori Error Estimates

By a similar selection to ^{[5, 6]}, we consider an auxiliary variable , then (1) can be split into a coupled system by:

(2)

In (2), by making an inner product for () and (), respectively, we use Green’s formula to obtain the mixed weak formulation: find:such that:

(3)

where .

We give a partition with mesh length , (is a positive integer) of the time interval . Furthermore, we define for a smooth function on. In the following, we will get two-step time discrete scheme based on the above expression.

At , we get an equivalent form of (3) as follows:

(4)

By two-step discrete method, (4) can be rewritten as:

(5)

where

(6)

Based on the weak formulation (5), the fully discrete two-step mixed scheme is to find in

such that

(7)

We now introduce two important lemmas with and to analyze some a priori error results

Lemma 1 ^{[10, 21]} Define a Ritz projection of by:

(8)

and the estimates hold

(9)

Lemma 2 ^[10] Let be the standard finite element interpolant of , then holds

(10)

For the need for error analysis, we decompose

Combine (5) and (7) with (8) to get the error equations:

(11)

and

(12)

Lemma 3 ^[15] For , and , the following errors hold:

(13)

(14)

(15)

Lemma 4 ^[15] For a sequenceand bounded function, the following equality holds

(16)

Theorem 1 Assuming that and , then, for hold

(17)

(18)

Proof: Set in (11) and use lemma 4 to get

(19)

By the application for Cauchy-Schwarz inequality and Young inequality, we arrive at

(20)

Multiply in (20), sum to and use lemma 3 to get

(21)

By an application for Poincare inequality, we have

(22)

We set in (12) to get

(23)

Apply Cauchy-Schwarz inequality and Young inequality to get

(24)

Use lemma 4 to get

(25)

Using differential mean value theorem, we have for and

(26)

As the result in ^[15], we have

(27)

Substitute (25)-(27) into (24) and note that lemma 3 to get

(28)

Multiply in (28), sum to and use (22) to get

(29)

By a combination of (9) and (10) with Gronwall lemma, we have

(30)

Substitute (30) into (21) and (22) to get

(31)

By a combination (28), (31), (9) and (10) with triangle inequality, we get the conclusion of theorem 1.

3. Numerical Test

In this section, for verifying the theoretical analysis, we take 1-D nonlinear pseudo-hyperbolic equation with space-time domain

(32)

where the exact solution is and is determined by the exact solution.

We choose the piecewise linear space in the spatial direction and two-step difference scheme in time direction and give some numerical results by Matlab procedure. In Table 1, we list some a priori error results for and in L²-norm with space-time step length . From Table 1, we can clearly see that the rate of convergence for our method is close to 2, which confirms our theoretical error results.

In Figure 1, the comparison figures are shown for the exact solution and the numerical solution . The similar results for and are also described in Figure 2. From Figure 1 and Figure 2, we can intuitively see that the proposed numerical method is feasible and effective.

Table 1. Optimal errors in L^∞ (L²) norm

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Figure 1. Exact solution u and numerical solution uⁿ

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Figure 2. Exact solution q and numerical solution qⁿ

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4. Some Conclusions and Remarks

So far, the H¹-Galerkin mixed element method ^[10] has been considered to seek for the numerical solutions by more and more people. In the literatures on the H¹-Galerkin mixed method, the backward Euler method and Crank-Nicolson scheme are widely presented. However, the two-step difference method based on the H¹-Galerkin mixed method is sparingly studied and analyzed. In this article, we consider a two-step difference method in time direction combining H¹-Galerkin mixed method and obtain some optimal time second-order rates of convergence.

From the results of numerical calculations, we can clearly find that fully discrete two-step H¹-Galerkin mixed method is efficient. In the near future, the fully discrete two-step difference method combined with H¹-Galerkin mixed method for nonlinear pseudo-hyperbolic integro-differential equations is analyzed.

Acknowledgement

In this paper, the research was sponsored by the National Natural Science Fund (Project No. 11301258; 11361035), Natural Science Fund of Inner Mongolia Autonomous Region (Project No. 2012MS0108; 2012MS0106), Scientific Research Projection of Higher Schools of (Project No. NJZZ12011; NJZY13199) and The National Undergraduate Innovative Training Project (201310126040).

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