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IMA Journal of Numerical Analysis Advance Access published online on June 19, 2009
IMA Journal of Numerical Analysis, doi:10.1093/imanum/drp004
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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.

Maximum-norm error analysis of a numerical solution via Laplace transformation and quadrature of a fractional-order evolution equation

William McLean{dagger}

School of Mathematics and Statistics, University of New South Wales, Sydney 2052, Australia

Vidar Thomée{ddagger} §

Department of Mathematics, Chalmers University of Technology, SE-41296 Göteborg, Sweden

{dagger} Corresponding author. Email: w.mclean{at}unsw.edu.au

{ddagger} Email: thomee{at}chalmers.se

Received on 29 April 2008. Accepted for publication 7 September 2008.


  Abstract

In a previous paper, McLean & Thomeé(2009, J. Integr. Equ. Appl. (to appear)), we studied three numerical methods for the discretization in time of a fractional-order evolution equation in a Banach space framework. Each of the methods applied a quadrature rule to a contour integral representation of the solution in the complex plane, where for each quadrature point an elliptic boundary-value problem had to be solved to determine the value of the integrand. The first two methods involved the Laplace transform of the forcing term, but the third did not. We analysed both the quadrature error and the error arising from a spatial discretization by finite elements, measured in the L2-norm. The present work extends our earlier results by proving error bounds in the technically more complicated case of the maximum norm. We also establish new regularity properties for the exact solution that are needed for our analysis.

Key Words: Laplace transform; quadrature; resolvent estimate; fractional diffusion

Dedicated to the memory of A. R. Mitchell, 1921–2007.

§ VT was a Visiting Professor at the University of New South Wales.


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