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

Abstract:

We consider an initial value problem for a class of evolution equations incorporating a memory term with a weakly singular kernel bounded by $C(t-s)^{\alpha -1}$, where $0<\alpha <1$. 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 $k^q+h^2\ell (k)$, where $k$ and $h$ are the mesh sizes in time and space, respectively, and $\ell (k)=\max (1,\log k^{-1})$. In the case $q=2$, we prove a higher convergence rate of order $k^3+h^2\ell (k)$ 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.