On the long-time stability of the implicit Euler scheme for the 2D space-periodic Navier–Stokes equations

In this article we discretize the two-dimensional space-periodic Navier–Stokes equations in time using the implicit Euler scheme and, with the aid of the discrete Gronwall lemma and the uniform discrete lemma, we prove that the scheme is H²-uniformly stable in time. Moreover, in a final remark, we describe how the above result can be extended to show that the implicit Euler scheme is uniformly stable with respect to the H³-norm, for all time.