Discontinuous Galerkin method for an evolution equation with a memory term of positive type

We consider an initial value problem for a class of evolution equations incorporating a memory term with a weakly singular kernel bounded by C(t − s)α−1, where 0 < α < 1. For the time discretization we apply the discontinuous Galerkin method using piecewise polynomials of degree at most q − 1, for q = 1 or 2. For the space discretization we use continuous piecewise-linear finite elements. The discrete solution satisfies an error bound of order kq + h2 (k), where k and h are the mesh sizes in time and space, respectively, and (k) = max(1, log k−1). In the case q = 2, we prove a higher convergence rate of order k3+h2 (k) at the nodes of the time mesh. Typically, the partial derivatives of the exact solution are singular at t = 0, necessitating the use of non-uniform time steps. We compare our theoretical error bounds with the results of numerical computations.