Semi-discrete finite element approximations for linear parabolic integro-differential equations with integrable kernels

In this paper we consider finite element methods for general parabolic integro-differential equations with integrable kernels. A new approach is taken, which allows us to derive optimal Lp, 2 ≤ p ≤ ∞, error estimates and superconvergence. The main advantage of our method is that the semi-discrete finite element approximations for linear equations, with both smooth and integrable kernels, can be treated in the same way without the introduction of the Ritz-Volterra projection; therefore, one can make full use of the results of finite element approximations for elliptic problems.