Negative norm error control for second-kind convolution Volterra equations (Q1977838)

The authors consider a piecewise constant finite element approximation to the convolution Volterra equation of the second kind, \(u(t)=f(t)+\int_0^t\Phi (t-s)u(s) ds\), a.e. in \(\mathcal J= [0,T]\). Here \(u\in L_p(\mathcal{J})\), \(f\in L_p(\mathcal{J})\), \(\Phi\in L_1(\mathcal{J})\) (\(1<p\leq \infty \)). 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 examples are given for problems in which \(\Phi \) is of a form typical in viscoelasticity theory.