Convergence and asymptotic stability of Galerkin methods for a partial differential equation with piecewise constant argument

The paper deals with the convergence and asymptotic stability of Galerkin methods for a partial differential equation with piecewise constant argument. The optimal convergence orders are obtained for the semidiscrete and full discrete (backward Euler) methods respectively. Both the discrete solutions are proved to be asymptotically stable under the condition that the analytical solution is asymptotically stable.