Time delay and finite differences for the non‐stationary non‐linear Navier–Stokes equations

In the present paper we use a time delay ϵ > 0 for an energy conserving approximation of the non‐linear term of the non‐stationary Navier–Stokes equations. We prove that the corresponding initial‐value problem (N ϵ ) in smoothly bounded domains G ⊆ ℝ 3 is well‐posed. We study a semidiscretized difference scheme for (N ϵ ) and prove convergence to optimal order in the Sobolev space H 2 ( G ). Passing to the limit ϵ→0 we show that the sequence of stabilized solutions has an accumulation point such that it solves the Navier–Stokes problem (N o ) in a weak sense (Hopf).