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 ofdegree at most q-1, for q= 1 or 2. For the space discretization we use continuous piecewise-linear nite elements. The discrete solution satises an error bound of order k^q+h²l(k), where k and h are the mesh sizes in time and space, respectively, and l (k) = max(1, log^-1). In the case q = 2, we prove a higher convergence rate oforder k³+h²l(k) at the nodes ofthe time mesh. Typically, the partial derivatives ofthe exact solution are singular at t=0, necessitating the use ofnon-uniform time steps. We compare our theoretical error bounds with the results ofnumerical computations.