Nonconforming Mixed Finite Element Method for Nonlinear Hyperbolic Equations

A nonconforming mixed finite element method for nonlinear hyperbolic equations is discussed. Existence and uniqueness of the solution to the discrete problem are proved. Priori estimates of optimal order are derived for both the displacement and the stress.


Introduction
In this paper, we discuss a nonconforming mixed finite element method for the following nonlinear hyperbolic initial and boundary value problem.
b u u a u u f u x y t T u X t g X t x y t T u x y u x y u x y u x y x y In order to describe the results briefly, we suppose that Equation (1) satisfy following assumptions on the data: 1) and are smooth and there exist constants and satisfying 2)         0 , , , , f u a u g X t u X and are sufficiently smooth functions with bounded derivatives.

  1 u X
There have been many very extensive studies about this kind of hyperbolic equations.For example, [1][2][3] studied the linear situations and gave error estimates under semi-discrete and fully-discrete schemes by standard Galerkin methods.[4,5] considered the mixed finite element methods for linear hyperbolic equations and obtained L 2 prior estimates about continuous time.In addition, [6] analyzed the mixed finite element methods for second order nonlinear hyperbolic equations.But all the above investigations are mainly about conforming situations and projections are indispensable.As we know, the nonconforming finite element methods arise because of the demands for reducing the calculation cost.[7] has pointed that the nonconforming finite element methods with degree of freedom defined on the element edges or element itself are appropriate for each degree of freedom belong to at most elements.
In the present work, we focus on the nonconforming mixed finite element approximation scheme for nonlinear hyperbolic equations.Firstly, we introduce the corresponding space and the interpolation operators.Secondly, Existence and uniqueness of the solutions to the discrete problem are proved.Finally, Priori estimates of optimal order are derived for both the displacement and the stress.
Throughout this paper, C denotes a general positive constant which is independent of , and K h is the diameter of the finite element K.

Construction of the Elements
Let h J be the a rectangular subdivision of 2 R   , and x y be the barycenter, the length of edges parallel to x-axis and y-axis by 2 x h and 2 y h .Then there exists an affine mapping ˆˆ: , , where x y plane and is the edges.We define the finite element ˆi l The interpolation functions defined above are properly and can be expressed as: , the associated finite element spaces as where   w denotes the jump of w across the boundary F, . , the interpolation operators: ˆˆ, ,

Main Results in Semi-Discrete Scheme
In this section, we will give the main results in this paper, including the existence and uniqueness of the solution to the discrete problem and priori estimates of optimal order.Firstly, we introduce and rewrite the Equation (1) as a system: and the norm by  , and let Thus the corresponding weak formulation of Equation ( 1) is to find a pair of   , where , d g w n gw n s


The semi-discrete mixed finite element procedure is determined: where , d

It can be seen that h
 and h  are the norms for and , respectively.
Then semi-discrete scheme can be rewritten as: Find , such that for every

A BE F
 and are Lipschitz continuous, it has a unique solution according to the theory of differential equations [8].
Thus, we complete the proof of Lemma 1. Lemma 2. [9] For Now we give the main result of this paper.
and  be the solutions of Equations ( 2) and (3), respectively.For , there hold that Proof: Firstly, by the interpolation condition and definition, it is easy to see that and . Secondly, for every h , v is a constant, by application of Green's formula and the interpo- and It is easy to see that h and satisfy the following error equations Using derivation about time t of Equation ( 4), combining Equation (5), we obtain Choosing h v   and h w   in Equation ( 6), and integrating from 0 to t, we have We will give the analysis result of Equation ( 7) in detail.Firstly, by the initial condition, it is followed that Secondly, by Cauchy-Schwartz's inequality and Young's inequality, we obtain Similarly, by the initial condition of     , f u b u and   a u , we use Young's inequality to get Substituting the above estimates, and applying Lemma 2, we get Finally, by the triangle inequality, we complete the proof.

Theorem 1 .
The above problem (3) has a unique solution.