Non-Classical Elliptic Projections and $L^2$-Error Estimates for Galerkin Methods for Parabolic Integro-Differential Equations

In this paper we shall define a so-called "non-classical" elliptic projection associated with an integro-differential operator. The properties of this projection will be analyzed and used to obtain the optimal $L^2$ error estimates for the continuous and discrete time Galerkin procedures when applied to linear integro-differential equations of parabolic type.