-Galerkin mixed finite element methods for pseudo-hyperbolic equations☆
Introduction
In this paper, we consider the following initial-boundary value problem of pseudo-hyperbolic systemwhere is a bounded convex polygonal domain in with Lipschitz continuous boundary is the time interval with . is smooth functions with bounded derivatives, and are given functions, andfor positive constants and .
The pseudo-hyperbolic equations are a high-order partial differential equations 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], Pao [2] and Arima and Hasegawa [3] gave some further study and extension for the numerical model. Ponce [4], [5] have given some results about the uniqueness and existence, and the asymptotic behavior of solutions for this problem. Hui and Hongxing [6] used two least-squares Galerkin finite element schemes to solve pseudo-hyperbolic equations. Moreover, the two methods get the approximate solutions with first-order and second-order accuracy in time increment, respectively.
An -Galerkin method involving -finite element spaces is discussed for parabolic problems in [8], [9]. Apart from difficulties in constructing such spaces, computationally more attractive and widely used piecewise linear elements are excluded from this class of finite dimensional spaces. In order to relax -smoothness and use -elements, we first split the problem into a first-order system and then propose a nonsymmetric version of a least square method that is an -Galerkin procedure for the solution and its flux.
In recent years, a lot of researchers have studied mixed finite element methods for partial differential equation. This method was initially introduced by engineers in the 1960’s [21], [22], [23] for solving problems in solid continua. Since then, it has been applied to many areas such as solid and fluid mechanics [10], [11]. However, this procedure has to satisfy LBB consistency condition on the approximating subspaces and restricts the choice of finite element spaces.
With the research and development of mixed finite element methods, Pani [12] (in 1998) proposed a new mixed finite element method called -Galerkin mixed finite element procedure which is applied to a mixed system in u and its flux q. Compared to standard mixed methods, the proposed methods have several attractive features. First, they are not subject to the LBB consistency condition. The finite element spaces (for approximating u) and (for approximating the flux q) may be of differing polynomial degrees. Moreover, the - and -error estimates do not require the finite element mesh to be quasi-uniform. Although we require extra regularity on the solution, a better order of convergence for the flux in -norm is obtained. From then on, the method was applied to the parabolic integro-differential equation [13] and hyperbolic problems [14] and so on. However, the convergence of -Galerkin mixed finite element method for the problem (1) has not been studied in the literature. In [12], [13], [14], optimal error estimates are derived for both semidiscrete and fully discrete schemes for problems in one space dimension. An extension to problems for two and three variables is also discussed. And some comparisons between classical mixed finite element methods and the -Galerkin mixed finite element methods are discussed. In [12], [14], a modified -Galerkin mixed finite element method is proposed to obtain optimal estimates for the stress in -norm without restricting the finite element space . In this paper, we propose the -Galerkin mixed finite element schemes for pseudo-hyperbolic equations. Depending on the physical quantities of interest, two methods are discussed. We get the optimal error estimates for both semidiscrete and completely discrete schemes in a single space variable. An extension to problems in two and three space variables is also discussed, and the proof of the existence and uniqueness of the discrete solutions is given. Moreover, the rates of convergence obtained coincide with those by Pani et al. [15], [20] using classical mixed method.
The layout of the paper is as follows. In Section 2, depending on the physical quantities of interest, two weak formulations of the -Galerkin mixed finite element are given, and optimal error estimates are given for semidiscrete schemes for problems in one space dimension. The fully discrete schemes are established and optimal error estimates are derived for fully discrete schemes for problems in a single space variable in Section 3. An extension to problems in two and three space variables is also discussed, and the proof of the existence and uniqueness of the discrete solutions is given in Section 4. Finally in Section 5, we will give some concluding remarks about -Galerkin mixed finite element.
Throughout this paper, C will denote a generic positive constant which does not depend on the spatial mesh parameter h and time-discretization parameter . At the same time, we give a important integral inequalitywhere is a integrable function in .
Section snippets
Semidiscrete scheme for problems in one space dimension
In this Section, we consider the following one dimensional second-order pseudo-hyperbolic equation with Dirichlet boundary conditions and initial conditionswhere .
Depend on the physical quantities of interest to consider two methods. Withwe reformulate the pseudo-hyperbolic Eq. (3) as the first-order system (I)If our concern is to
Error estimates of fully discrete schemes
In this section, we get the error estimates of fully discrete schemes. For the backward Euler procedure, let be a given partition of the time interval with step length , for some positive integer M. For a smooth function on , define and . Let and , respectively, be the approximations of u and q at which we shall define through the following scheme. Given in , we now determine a pair in
-Galerkin mixed finite method for several spaces variables
In this section, we consider the several spaces variablesLet , ( or 3) with inner product and normFurther, let with the usual inner product and norm. Letwith normSetting , we haveWe define the semidiscrete -Galerkin mixed finite
Concluding remarks
In the paper, we get both semidiscrete and fully discrete priori error estimates of -Galerkin mixed finite element method for a class of second-order pseudo-hyperbolic equations in a single space variables. An extension of system (I) is given for the problems in two and three space variables in Section 4, and the Galerkin approximation has the same rate of convergence as in the case of classical mixed finite element method, but without LBB consistency condition and also without quasi-uniform
Acknowledgement
The authors thank the referees for their valuable comments and suggestions.
References (23)
A mixed initial boundary value problem arising in neurophysiology
J. Math. Anal. Appl.
(1975)Global existence of small of solutions to a class of nonlinear evolution equations
Nonlinear Anal.
(1985)- et al.
Least-squares Galerkin procedures for pseudo-hyperbolic equations
Appl. Math. Comput.
(2007) - et al.
Finite element methods for nonlinear Sobolev equations with nonlinear boundary conditions
J. Math. Anal. Appl.
(1992) - et al.
An active pulse transmission line simulating nerve axon
Proc. IRE
(1962) - et al.
On global solutions for mixed problems of a semi-linear differential equation
Proc. Jpn. Acad.
(1963) - et al.
Long time behaviors of solutions for initial boundary value problem of pseudohyperbolic equations
Acta Math. Appl. Sin.
(1999) - et al.
Non-classical projection and Galerkin methods for nonlinear integro-differential equations
Calcolo.
(1988) - et al.
-Galerkin methods for the Laplace and heat equations
- et al.
An -Galerkin method for quasilinear parabolic partial differential equations
A time-discretization procedure for a mixed finite element approximation of miscible displacement in porous media
RAIRO Anal. Numér.
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Supported by National Natural Science Fund No. 10601022, NSF of Inner Mongolia Autonomous Region No. 200607010106, YSF of Inner Mongolia University No. ND0702.