A Priori Error Analysis and Numerical Simulation of Fully Discrete H1-Galerkin Mixed Element Method for Nonlinear Pseudo-Hyperbolic Equation
1School of Mathematical Sciences, Inner Mongolia University, Hohhot, China
In this article, a fully discrete two-step H1-Galerkin mixed method is presented for nonlinear pseudo-hyperbolic equation. The spatial direction and time direction are approximated by H1-Galerkin mixed method and two-step difference method, respectively. Some a priori error results are analyzed for the scalar unknown function u and its flux . Moreover, a numerical test is made to verify our theoretical error analysis.
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Keywords: two-step discrete method, H1-Galerkin mixed method, nonlinear pseudo-hyperbolic equation, a priori error analysis, fully discrete scheme
American Journal of Numerical Analysis, 2014 2 (2),
Received January 26, 2014; Revised February 20, 2014; Accepted February 28, 2014Copyright © 2014 Science and Education Publishing. All Rights Reserved.
Cite this article:
- Zhang, Guoyu, et al. "A Priori Error Analysis and Numerical Simulation of Fully Discrete H1-Galerkin Mixed Element Method for Nonlinear Pseudo-Hyperbolic Equation." American Journal of Numerical Analysis 2.2 (2014): 55-59.
- Zhang, G. , Huang, L. , Yu, Z. , Liu, Y. , Li, H. , & Zhang, M. (2014). A Priori Error Analysis and Numerical Simulation of Fully Discrete H1-Galerkin Mixed Element Method for Nonlinear Pseudo-Hyperbolic Equation. American Journal of Numerical Analysis, 2(2), 55-59.
- Zhang, Guoyu, Lijun Huang, Zudeng Yu, Yang Liu, Hong Li, and Min Zhang. "A Priori Error Analysis and Numerical Simulation of Fully Discrete H1-Galerkin Mixed Element Method for Nonlinear Pseudo-Hyperbolic Equation." American Journal of Numerical Analysis 2, no. 2 (2014): 55-59.
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In this article, we consider the following nonlinear pseudo-hyperbolic equation [1-9] with initial and boundary conditions
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  and so forth.
Pani  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 , hyperbolic equation , Sobolev equation , RLW equation [14, 15], Schrodinger equation , Burgers’ equation ) were formulated. At the same time, some good numerical methods (Nonconforming H1-Galerkin mixed method [7, 17], Splitting H1-Galerkin mixed method , H1-Galerkin expanded mixed methods [18, 19]) based on the H1-Galerkin mixed element method were developed by many authors. Liu and Li , Zhou  gave the theoretical analysis of a priori errors for H1-Galerkin mixed method, respectively. Zhang et al.  studied a semi-discrete nonconforming H1-Galerkin mixed scheme. Liu et al.  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 Estimates2.1. Mixed Weak Formulation
In (2), by making an inner product for () and (), respectively, we use Green’s formula to obtain the mixed weak formulation: find:such that:
where .2.2. Fully Discrete Scheme and Some Important Lemmas
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:
By two-step discrete method, (4) can be rewritten as:
Based on the weak formulation (5), the fully discrete two-step mixed scheme is to find in
We now introduce two important lemmas with and to analyze some a priori error results
and the estimates hold
Lemma 2  Let be the standard finite element interpolant of , then holds
For the need for error analysis, we decompose
Combine (5) and (7) with (8) to get the error equations:
Lemma 3  For , and , the following errors hold:
Lemma 4  For a sequenceand bounded function, the following equality holds
Theorem 1 Assuming that and , then, for hold
Proof: Set in (11) and use lemma 4 to get
By the application for Cauchy-Schwarz inequality and Young inequality, we arrive at
Multiply in (20), sum to and use lemma 3 to get
By an application for Poincare inequality, we have
We set in (12) to get
Apply Cauchy-Schwarz inequality and Young inequality to get
Use lemma 4 to get
Using differential mean value theorem, we have for and
As the result in , we have
Substitute (25)-(27) into (24) and note that lemma 3 to get
Multiply in (28), sum to and use (22) to get
By a combination of (9) and (10) with Gronwall lemma, we have
Substitute (30) into (21) and (22) to get
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
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.
4. Some Conclusions and Remarks
So far, the H1-Galerkin mixed element method  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.
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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