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

In a previous paper, McLean & Thomee (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 L(2)-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.