Ritz-Volterra reconstructions and a posteriori error analysis of finite element method for parabolic integro-differential equations

We derive a posteriori error estimates for both semidiscrete and implicit fully discrete backward Euler method for linear parabolic integro-differential equations in a bounded convex polygonal or polyhedral domain. A novel space-time reconstruction operator is introduced, which is a generalization of the elliptic reconstruction operator [2003, SIAM J. Numer. Anal., 41, pp. 1585-1594], and we call it as Ritz-Volterra reconstruction operator. The Ritz-Volterra reconstruction operator in conjunction with the linear approximation of the Volterra integral term is used in a crucial way to derive optimal order a posteriori error estimates in L-az(L-2) and L-2(H-1)-norms. The related a posteriori error estimates for the Ritz-Volterra reconstruction error are also established. We allow only nested refinement of the space meshes for the fully discrete analysis.