The numerical solution of partial integodifferential equations (with homogeneous Dirichlet boundary conditions and given initial values) by time-continuous finite-element methods is considered. The paper presents convergence results based on the decomposition \(u_ h-u=(u_ h-V_ hu)+(V_ hu-u)\) of the error, where \(V_ h\) is the so-called Ritz- Volterra projection.
First, various error estimates for the Ritz-Volterra projection (in \(L_ p\) for \(2\leq p\leq \infty)\) are given. Separate sections are then devoted to their application to parabolic and hyperbolic integrodifferential equations, and to Sobolev and viscoelasticity type equations.