A numerical test of the reliability of galerkin approximations to the solutions of the Navier-Stokes equation

The method of Galerkin appears to provide an attractive approach for the approximate solution of the Navier-Stokes equation. However, there can be difficulties in connection with the selection of the set of Fourier modes and the correct behavior of the Galerkin approximation. A description is, therefore, provided for a test which can be used to decide a posteriori if the exact solution behaves like the numerically computed Galerkin approximation. In the derivation of the basic relations for the test, attention is given to the two-dimensional flow of an incompressible viscous fluid with unit density contained in a finite volume with rigid boundary and governed by the Navier-Stokes equation.