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IMA Journal of Numerical Analysis Advance Access published online on February 26, 2009
IMA Journal of Numerical Analysis, doi:10.1093/imanum/drn075
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© The author 2009. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

A second-order accurate numerical method for a semilinear integro-differential equation with a weakly singular kernel

Kassem Mustapha{dagger}

Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia

Hussein Mustapha{ddagger}

Department of Mining and Materials Engineering, McGill University, Montreal, Canada

{dagger} Corresponding author. Email: kassem{at}kfupm.edu.sa

{ddagger} Email: hussein.mustapha{at}mcgill.ca

Received on 30 August 2007. Revised on 1 May 2008.


  Abstract

We study a generalized extrapolated Crank–Nicolson scheme for the time discretization of a semilinear integro-differential equation with a weakly singular kernel, in combination with a space discretization by linear finite elements. The scheme uses variable grids in time to compensate for the singular behaviour of the exact solution at t = 0. With appropriate assumptions on the data and assuming that the spatial domain is convex or smooth, we show that the error is of order k2 + h2, where k and h are the parameters for the time and space meshes, respectively. The results of numerical computations demonstrate the convergence of our scheme.

Key Words: integro-differential equation; weakly singular kernel; nonuniform time steps; quadrature error; finite elements; Gronwall's lemma


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