Parametric elliptic reconstructions and a posteriori error estimates for parabolic partial differential equations with small randomness in a Robin boundary condition

Abstract

We obtain reliable a posteriori residual-based error estimates for parabolic partial differential equations with small randomness in a Robin-type boundary condition. The uncertainty is addressed via a perturbation approach, transforming the problem with small random input data into a sequence of deterministic problems. Finite element approximations, combined with the backward Euler time discretization, are employed for the resulting problems. To ensure optimal spatial accuracy, the elliptic reconstruction framework is suitably adapted to this setting. This is achieved by introducing a parametric elliptic reconstruction operator that unifies the a posteriori analysis of deterministic parabolic problems with that of parabolic problems with small uncertainties. The obtained a posteriori error estimator is robust with respect to the parameter that describes the amount of uncertainty, in the sense that the constants appearing in the bounds are independent of the parameter, as well as of the mesh size and the time-step. In addition, numerical experiments are presented to validate the theoretical findings and illustrate the robustness of the proposed estimators.