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In this paper, both semidiscrete and completely discrete finite volumeelement methods (FVEMs) are analyzed for approximating solutions of a class oflinear hyperbolic integro- differential equations in a two-dimensional convexpolygonal domain. The effect of numerical quadrature is also examined. In thesemidiscrete case, optimal error estimates in L^{\infty}(L2) andL^{\infty}(H1)- norms are shown to hold with minimal regularity assumptions onthe initial data, whereas quasi-optimal estimate in derived inL^{\infty}(L^{\infty})-norm under higher regularity on the data. Based on asecond order explicit method in time, a completely discrete scheme is examinedand optimal error estimates are established with a mild condition on the spaceand time discretizing parameters. Finally, some numerical experiments areconducted which confirm the theoretical order of convergence.

A priori error estimates for finite volume element approximations to second order linear hyperbolic integro-differential equations