Adaptive Error Control

The derivation of reliable and efficient a posteriori error estimates for the Navier-Stokes equations is an important consideration in computational fluid dynamics. On perusing existing a priori error estimates for the Navier-Stokes equations, one notices fundamental differences from and fresh difficulties compared with the estimates for diffusion-dominated and convection-dominated elliptic problems in Chapter II.3.6. Some new obstacles that appear are

• a smallness condition on the Reynolds number Re = 1/v to ensure uniqueness of the solution (Chapter 1)

• a well-posedness assumption for a linearized problem with an a priori error estimate that depends strongly on an unknown stability constant (Theorem 3.14).

As we saw in Example 3.16, this stability constant can grow exponentially in the Reynolds number Re. One should be aware that such a property will restrict considerably the quantitative value of our estimates. On the other hand, in the special case of a no-flow problem (see Remark 2.10), one has uniqueness of the solution for all Reynolds numbers and error estimates with a right-hand side that is a polynomial function of Re.