A priori error estimates for finite volume element approximations to   second order linear hyperbolic integro-differential equations
  In this paper, both semidiscrete and completely discrete finite volume element methods (FVEMs) are analyzed for approximating solutions of a class of linear hyperbolic integro- differential equations in a two-dimensional convex polygonal domain. The effect of numerical quadrature is also examined. In the semidiscrete case, optimal error estimates in L^{\infty}(L2) and L^{\infty}(H1)- norms are shown to hold with minimal regularity assumptions on the initial data, whereas quasi-optimal estimate in derived in L^{\infty}(L^{\infty})-norm under higher regularity on the data. Based on a second order explicit method in time, a completely discrete scheme is examined and optimal error estimates are established with a mild condition on the space and time discretizing parameters. Finally, some numerical experiments are conducted which confirm the theoretical order of convergence. 
Samir Karaa and Amiya K. Pani
math.NA