Superconvergence of a Finite Element Method for Linear Integro-differential Problems

We introduce a new way of approximating initial condition to the semidiscrete finite element method for integro-differential equations using any degree of elements. We obtain several superconvergence results for the error between the approximate solution and the Ritz-Volterra projection of the exact solution. For k > 1, we obtain first order gain in Lp(2≤ p ≤∞) norm, second order in W1,p(2≤ p ≤∞) norm and almost second order in W1,∞ norm. For k = 1, we obtain first order gain in W1,p(2 ≤ p ≤ ∞) norms. Further, applying interpolated postprocessing technique to the approximate solution, we get one order global superconvergence between the exact solution and the interpolation of the approximate solution in the Lp and W1,p(2≤ p ≤∞).