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

In a previous paper, McLean and Thomée (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, needed for our analysis.