Finite element and DG analysis for neutral-type Volterra integro-differential equations with weakly singular kernels

In this paper we analyze the numerical solution of Volterra integro-differential equations of neutral type with weakly singular kernels. We establish a priori error estimations for the finite-element-method semi-discretization of the given problem by defining a suitable Ritz–Volterra projection operator: here, the key point in the proof is the fact that the definition of the Ritz–Volterra projection operator that is not related to the neutral term. We then discuss the discontinuous Galerkin time-stepping method for the semi-discretized equation, together with a fully discretized form.