The effect of spatial quadrature on finite element Galerkin approximations to hyperbolic integro-differential equations

Abstract

The purpose of this paper is to study the effect of numerical quadrature on the finite element approximations to the solutions of hyperbolic integro-differential equations. Both semidiscrete and fully discrete schemes are analyzed and optimal estimates are derived in L-infinity(H-1), L-infinity(L-2) norms and quasi-optimal estimate in L-infinity(L-infinity) norm using energy arguments. Further, optimal L-infinity(L-2)-estimates are shown to hold with minimal smoothness assumptions on the initial functions. The analysis in the present paper not only improves upon the earlier results of Baker and Dougalis [SIAM J. Numer. Anal. 13 (1976), pp. 577-598] but also confirms the minimum smoothness assumptions of Rauch [SIAM J. Numer. Anal. 22 (1985), pp. 245-249] for purely second order hyperbolic equation with quadrature.