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

TL;DR

This paper develops and analyzes finite volume element methods for second order linear hyperbolic integro-differential equations, providing optimal error estimates and confirming them through numerical experiments.

Contribution

It introduces new error estimates for FVEMs applied to hyperbolic integro-differential equations, including effects of numerical quadrature and discrete schemes.

Findings

Optimal error estimates in various norms are established.

Numerical experiments confirm theoretical convergence orders.

The analysis covers both semidiscrete and fully discrete schemes.

Abstract

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.