H1-Galerkin mixed finite element methods for pseudo-hyperbolic equations

https://doi.org/10.1016/j.amc.2009.02.039Get rights and content

Abstract

H1-Galerkin mixed finite element methods are discussed for a class of second-order pseudo-hyperbolic equations. Depending on the physical quantities of interest, two methods are discussed. Optimal error estimates are derived for both semidiscrete and fully discrete schemes for problems in one space dimension. An extension to problems in two and three space variables is also discussed, the existence and uniqueness are derived and it is showed that the H1-Galerkin mixed finite element approximations have the same rate of convergence as in the classical mixed finite element methods without requiring the LBB consistency condition.

Introduction

In this paper, we consider the following initial-boundary value problem of pseudo-hyperbolic systemutt-·(a(x)ut+a(x)u)+ut=f(x,t),(x,t)Ω×J,u(x,t)=0,(x,t)Ω×J¯,u(x,0)=u0(x),ut(x,0)=u1(x),xΩ.where Ω is a bounded convex polygonal domain in Rd(d=1,2,3) with Lipschitz continuous boundary Ω,J=(0,T] is the time interval with 0<T<. a(x) is smooth functions with bounded derivatives, f(x,t),u0(x) and u1(x) are given functions, and0<amina(x)amax,xΩ,for positive constants amin and amax.

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 H1-Galerkin method involving C1-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 C1-smoothness and use C0-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 H1-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 H1-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 Vh (for approximating u) and Wh (for approximating the flux q) may be of differing polynomial degrees. Moreover, the L2- and H1-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 L2-norm is obtained. From then on, the method was applied to the parabolic integro-differential equation [13] and hyperbolic problems [14] (utt-·(a(x)u)+c(x)u=f(x,t)) and so on. However, the convergence of H1-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 H1-Galerkin mixed finite element methods are discussed. In [12], [14], a modified H1-Galerkin mixed finite element method is proposed to obtain optimal estimates for the stress in L(L2(Ω))-norm without restricting the finite element space Wh. In this paper, we propose the H1-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 H1-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 H1-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 Δt. At the same time, we give a important integral inequality0t0τ|ψ(s)|2dsdτC0t|ψ(s)|2ds.where ψ is a integrable function in [0,t],t[0,T].

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 conditionsutt-(a(x)utx+a(x)ux)x+ut=f(x,t),(x,t)Ω×J,u(0,t)=u(1,t)=0,tJ¯,u(x,0)=u0(x),ut(x,0)=u1(x)xΩ.where Ω=(0,1),J=(0,T].

Depend on the physical quantities of interest to consider two methods. Withq=a(x)utx+a(x)ux,we reformulate the pseudo-hyperbolic Eq. (3) as the first-order system (I)a(x)utx+a(x)ux=q,utt-qx+ut=f(x,t).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 0=t0<t1<t2<<tM=T be a given partition of the time interval [0,T] with step length tn=nΔt,Δt=T/M, for some positive integer M. For a smooth function ϕ on [0,T], define ϕn=ϕ(tn) and ¯tϕn=(ϕn-ϕn-1)/Δt. Let Un and Zn, respectively, be the approximations of u and q at t=tn which we shall define through the following scheme. Given {Un-1,Zn-1} in Vh×Wh, we now determine a pair {Un,Zn} in Vh×Wh

H1-Galerkin mixed finite method for several spaces variables

In this section, we consider the several spaces variablesutt-·(a(x)ut+a(x)u)+ut=f(x,t),(x,t)Ω×J,u(x,t)=0,(x,t)Ω×J¯,u(x,0)=u0(x),ut(x,0)=u1(x)xΩ.Let L2(Ω)=(L2(Ω))d, (d=2 or 3) with inner product and norm(q,w)=i=1d(qi,wi)andw=i=1dwi212.Further, let Hm=(Hm(Ω))d with the usual inner product and norm. LetW=H(div;Ω)={wL2(Ω)|·wL2(Ω)},with normwH(div;Ω)=(w2+·w2)12.Setting q=a(x)ut+a(x)u, we haveut+u=αq,utt-·q+ut=f(x,t).We define the semidiscrete H1-Galerkin mixed finite

Concluding remarks

In the paper, we get both semidiscrete and fully discrete priori error estimates of H1-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.

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