A Posteriori Error Estimates for the Crank–Nicolson Method: Application to Parabolic Partial Differential Equations Subject to a Robin Boundary Condition with Small Randomness

In this article, we obtain residual-based a posteriori error estimates for a linear parabolic partial differential equation which is subject to a Robin boundary condition that contains a small uncertainty. To this end, the perturbation technique is exploited to express the exact random solution in terms of a power series with respect to the uncertainty parameter, whence we obtain deterministic problems. Each problem is then discretized in space by continuous piecewise linear elements, and the Crank–Nicolson scheme is used for time-stepping. Reconstruction techniques are employed to obtain optimal bounds. Numerical investigations are performed that confirm the theoretical findings.