Discontinuous Galerkin method with a posteriori$L_p(0,t_i)$ error estimate for second-kind Volterra problems

Summary.

We consider a discontinuous Galerkin finite element method applied in time to a model Volterra equation of the second kind. A residual-based computable$L_p(0,t_i)$ Galerkin-error estimate is derived for $1 \le p \le \infty$. This estimate does not explicitly contain the time step and therefore the time step control must be based on a heuristic criterion, the estimate can then be used to demonstrate the integrity, or otherwise, of the finite element solution. After performing some numerical experiments we conclude that this approach is at least competetive with classical discretizations since it is computationally simple to implement, but has the added advantage of reliable error feedback.