Delay-dependent elliptic reconstruction and optimal 𝐿^{∞}(𝐿²) a posteriori error estimates for fully discrete delay parabolic problems

Abstract

We derive optimal order a posteriori error estimates for fully discrete approximations of linear parabolic delay differential equations (PDDEs), in the L ∞ ( L 2 ) L^\infty (L^2) -norm. For the discretization in time we use Backward Euler and Crank-Nicolson methods, while for the space discretization we use standard conforming finite element methods. A novel space-time reconstruction operator is introduced, which is a generalization of the elliptic reconstruction operator, and we call it as delay-dependent elliptic reconstruction operator. The related a posteriori error estimates for the delay-dependent elliptic reconstruction play key roles in deriving optimal order a posteriori error estimates in the L ∞ ( L 2 ) L^\infty (L^2) -norm. Numerical experiments verify and complement our theoretical results.