Negative norm error control for second-kind convolution Volterra equations

We consider a piecewise constant finite element approximation to the convol ution Volterra equation problem of the second kind: find u such that u = f + phi * u in a time interval [0, T]. An a posteriori estimate of the error measured in the W-p(-1) (0,T) norm is developed and used to provide a time step selection criterion for an adaptive solution algorithm. Numerical exam ples are given for problems in which phi is of a form typical in viscoelast icity theory, Mathematics Subject Classification (1991): 3F15, 45D05, 45K05 , 65M15.