Fully discrete a posteriori error estimates for parabolic integro-differential equations using the two-step backward differentiation formula

Abstract

In this article, finite element a posteriori error estimates for the linear parabolic integro-differential equation using the two-step backward time descretization formula are explored. For space discretization, we use piecewise linear finite element spaces. The Ritz–Volterra reconstruction operator is used as a raw ingredient to obtain the optimal convergence in space. Further, a novel quadratic space–time reconstruction operator, namely BDF2 operator, is introduced to achieve second-order accuracy in time. Numerical results demonstrate the theoretical findings.