TY - JOUR T1 - A Priori Error Estimates for Finite Volume Element Approximations to Second Order Linear Hyperbolic Integro-Differential Equations JO - International Journal of Numerical Analysis and Modeling VL - 3 SP - 401 EP - 429 PY - 2015 DA - 2015/12 SN - 12 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/496.html KW - Finite volume element, hyperbolic integro-differential equation, semidiscrete method, numerical quadrature, Ritz-Volterra projection, completely discrete scheme, optimal error estimates. AB -
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 $L^∞(L^2)$ and $L^∞(H^1)$ norms are shown to hold with minimal regularity assumptions on the initial data, whereas quasi-optimal estimate is derived in $L^∞(L^∞)$ 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.