Nonconforming H 1 -galerkin Mixed Finite Element Method for Pseudo-hyperbolic Equations

Based on H 1-Galerkin mixed finite element method with nonconforming quasi-Wilson element, a numerical approximate scheme is established for pseudo-hyperbolic equations under arbitrary quadrilateral meshes. The corresponding optimal order error estimate is derived by the interpolation technique instead of the generalized elliptic projection which is necessary for classical error estimates of finite element analysis.


 
a X is smooth function with bounded derivatives,   0 u X , and f are given functions, and for positive constants and .
min max The pseudo-hyperbolic equation is a high-order partial differential system with mixed partial derivative with respect to time and space, which describe heat and mass transfer, reaction-diffusion and nerve conduction, and other physical phenomena.This model was proposed by Nagumo et al. [1].Wan and Liu [2] have given some results about the asymptotic behavior of solutions for this problem.Guo and Rui [3] used two least-squares Galerkin finite element schemes to solve pseudo-hyperbolic equations.

a a
On the other hand, H 1 -Galerkin mixed finite element method (see [4]) has been under rapid progress recently since this method has the following advantages over classical mixed finite element method.The method allows the approximation spaces to be polynomial spaces with different orders without LBB consistency condition and there is no requirement of the quasi-uniform assumption on the meshes.For example, Pani [4,5] proposed an H 1 -Galerkin mixed finite element procedure to deal with parabolic partial differential equations and parabolic partial integro-differential equations, respectively.Liu and Li [6,7] applied this method to deal with pseudohyperbolic equations and fourth-order heavy damping wave equation.Further, Shi and Wang [8] investigated this method for integro-differential equation of parabolic type with nonconforming finite elements including the ones studied in [9,10].
It is well-known that the convergence behavior of the well-known nonconforming Wilson element is much better than that of conforming bilinear element.So it is widely used in engineering computations.However, it is only convergent for rectangular and parallelogram meshes.The convergence for arbitrary quadrilateral meshes can not be ensured since it passes neither Irons Patch Test [11] nor General Patch Test [12].In order to extend this element to arbitrary quadrilateral meshes, various improved methods have been developed in [13][14][15][16][17][18][19][20][21][22][23][24].In particular, [19][20][21][22][23][24] generalized the results mentioned above and constructed a class of Quasi-Wilson elements which are convergent to the second order elliptic problem for narrow quadrilateral meshes [23].
In the present work, we will focus on H 1 -Galerkin nonconforming mixed finite element approximation to problem (1) under arbitrary quadrilateral meshes.We firstly prove the existence and uniqueness of the solution for semi-discrete scheme.Then, based on a very special property of the quasi-Wilson element i.e. the consistency error is one order higher than interpolation error, we deduce the optimal order error estimates for semidiscrete scheme directly without using the generalized elliptic projection which is a indispensable tool in the tradition finite element methods.
This paper is arranged as follows.In Section 2, we briefly introduce the construction of nonconforming mixed finite element.In section III, we will discuss the H 1 -Galerkin mixed finite element scheme for pseudohyperbolic equations.At last, the corresponding optimal order error estimates are obtained for semi-discrete scheme.

Construction of Nonconforming Mixed Finite Element
Assume 4 ˆˆl a a  1 be the four edges of K .
We define the finite elements by ˆˆˆˆˆˆ, , , , , p p p p p p   where , , ,   , it is the so-called Wilson ele- ment.The interpolations defined above are properly posed and the interpolation functions can be expressed as satisfies the regularity assumption [11], where K denotes a convex quadrilateral with vertices Then there exists a invertible mapping : The associated finite element space and are defined as Then for all , we define the interpolation operators and L  be the set of square integrable functions on  and the space of two dimensional vectors which have all components in with its norm Throughout this paper, C denotes a general positive constant which is independent of h.

Nonconforming H 1 -Galerkin Mixed Finite Element Method for the Semi-Discrete Scheme
, then the corresponding weak formulation is: Find The corresponding semi-discrete finite element pro- For all , we define , It is easy to see that h  and  ; where , , , , , Sine (4) gives a system of nonlinear ordinary differential equations (ODEs) for the vector function , by the assumptions on   a X and the theory of ODEs, it follows that tion for 0 t  (see [25]).Ther the proof is complete.

Error Estimates
s the e solu efore In order to get the error which will play an impor estimates the following lemma tant role in our analysis and can be found in [24].
Lemma 1.For all where denotes the outward unit norm vector to n al K  .Now e will state the following main result of this er.
, w pap tions of the (2) and ( 3 and where It is easy to see that for all h , there hold the following error equations Schwartz's inequality yields in (7(a)) and using the Cauchy- For the right side of (9), applying  -Young's inequality, we have Finally, by use of the triangle inequality, ( 14) and ( 15), we get ( 5) and ( 6).The proof is completed.
For our subsequent use, we also use the standard sobolve space Integrating the both sides of (13) with respect to time from 0 to t, by Gronwall's lemma and noting