On Maximum Norm Estimates for Ritz Volterra Projection With Applications to Some Time Dependent Problems

The stability in L norm is considered for the Ritz Volterra projection and some applications are presented in this paper As a result point wise error estimates are established for the nite ele ment approximation for the parabolic integro di erential equation Sobolev equations and a di usion equation with non local boundary value problem This work is supported in part by NSERC CANADA Journal of Computational Mathematics Vol No x Introduction We are concerned with the nite element method for parabolic integro di erential equation ut t V t u t f t t T u v where V t is in general an integro di erential operator de ned on a Hilbert space X and that u and f are X valued functions de ned on J T with a positive time T A typical example of the Hilbert space X in the application will be the Sobolev space H consisting of functions de ned on an open bounded domain with vanished boundary value and rst order derivatives summable in L while the operator V t is the one de ned by V t u t A t u t Z t B t u d in for any u t H
where A t is a linear elliptic operator of second order and that B t any linear operator of no more than second order Although more examples of integro di erential operators will be considered in this paper we shall illustrate our results for the operator V t de ned by since the others can be modi ed to t the strategy designed for Numerical methods to the equation have been studied by several authors recently For nite di erence schemes we refer to and the refences cited therein The nite element method for this problem has also been studied in both smooth and non smooth data cases were considered and optimal error estimates in L were obtained the semi linear equation with non smooth data and an operator B of zero order was treated in along with a particular attention paid to the computation of the memory term by the quadrature rule Recently a di erent approach to the error analysis was proposed in and Their idea can be summerized as introducing a so called Ritz Volterra projection to decompose the error A systematic study of Ritz Volterra projection and its applications to parabolic and hyperbolic integro di erential equations Sobolev equation and the equations of visco elasticity can be found from For the sake of convenience of the analysis we shall take to be a plane convex polygonal domain Let Th be a quasi uniform triangulation so that h K ThK Let Sh be the nite element subspace associated with Th Without loss of generality we shall assume that Sh is made up of piece wise linear functions The object of this paper is to study the convergence behavior of the nite element approximation in the L norm As a matter of fact this problem had been considered by Lin and Zhang where an optimal maximum norm has been obtained for piecewise linear elements for a very special case that is the operators A and B are divergence form which allows us to use the standard regularized Green function and by Lin Thomee and Wahlbin in where the following estimate for any small ku t uh t k C u h r was derived based on their estimate in Lp Here r is the optimal order in the approximation and C u a constant dependent upon the exact solution u only It is clear that such an estimate is not optimal in compare with the results for the elliptic and parabolic equations cf We shall therefore study this problem from a di erent point of view in order to get a sharp estimate in the L norm The main idea of our approach can be summarized as rstly introducing an auxiliary problem associated with the Ritz Volterra operator V and then establishing our main results with the help of the solution of this auxiliary problem The auxiliary problem to be introduced in next section is an analogy of the regularized Green s function in the study of the L stability for the elliptic equation of second order Thus the only contribution of the authors would be to apply the known technique appropriately to the current problem However such an extension is not trivial due to the memory term involved in the operator V Our main result regards to the maximum norm error estimate for the Ritz Volterra projection Vh de ned by V t Vhu t V t u t Sh for each t J where V t is the bilinear form associated with the Ritz Volterra operator V t de ned by V t u t v t A t u t v t Z t B t u v t d for u t v t H
with t J Applications to nite element approximations for the parabolic integro di erential equation Sobolev equation and a di usion equation with non local boundary condition are presented in this paper This paper is organized as follows In x we shall introduce and study an auxiliary problem associated with the operator V The solution of this problem can be regarded as a certain regularized Green s function associated with the Ritz Volterra operator In x we shall establish an estimate in the L norm for the Ritz Volterra projection onto the nite element subspace Sh while the applications to the parabolic integro di erential equation Sobolev equation and a di usion equation with non local boundary condition will be given in x A preliminary of this paper can be summarized as follows Denote by Wm p the Sobolev space on the domain de ned by W p fv Dv L with jjj mg kvkm p m X