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

Abstract

Parabolic integro-differential equations (PIDE) arise in various applications. Some of its occurrence includes heat conduction in material with memory, the compression of poro-viscoelasticity media, nuclear reactor dynamics and the epidemic phenomena in biology. Despite being so rich in the a priori analysis and in spite of the importance of these equations in the modelling of several physical phenomena, the topic of a posteriori error analysis for such kind of equations remains unexplored. Since PIDE may be thought of as a perturbation of the purely parabolic problem, therefore, it is natural to see whether the a posteriori error analysis of parabolic problems can be extended to PIDE. An attempt has been made in this work to generalize the results of purely parabolic problems to PIDE. A posteriori error estimates for both semidiscrete and implicit fully discrete backward Euler method for linear parabolic integro-differential equations are obtained 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 [SIAM J. Numer. Anal., 41, pp. 1585–1594, (2003)], 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∞(L2) and L2(H1)-norms. The related a posteriori error estimates for the Ritz-Volterra reconstruction error are also established. It is observed that the Ritz–Volterra projection is useful in a priori analysis for a wide range of (linear and nonlinear) parabolic and hyperbolic integro-differential problems. We strongly believe that, the Ritz–Volterra reconstruction operator, a counterpart of the Ritz–Volterra projection in the a priori analysis, can be appropriately modified to obtain estimators for a class of integro-differential problems. Moreover, Ritz Volterra reconstruction operator unifies a posteriori approach from parabolic problems to PIDE.