Abstract.A semidiscrete finite volume element (FVE) approximation to a parabolic integrodifferential equation (PIDE) is analyzed in a two-dimensional convex polygonal domain. An optimalorder L 2 -error estimate for smooth initial data and nearly the same optimal-order L 2 -error estimate for nonsmooth initial data are obtained. More precisely, for homogeneous equations, an elementary energy technique and a duality argument are used to derive an error estimate of order O t −1 h 2 ln h in the L 2 -norm for positive time when the given initial function is only in L 2 .
In this article, the effect of numerical quadrature on the finite element Galerkin approximations to the solution of hyperbolic equations has been studied. Both semidiscrete and fully discrete schemes are analyzed and optimal estimates are derived in the
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