On solenoidal high-degree polynomial approximations to solutions of the stationary Navier–Stokes equations

The cell discretization algorithm, a nonconforming extension of the finite element method, is used to obtain approximations to the Navier–Stokes equations for viscous steady-state incompressible flow. An implementation using polynomial bases is described that permits the use of approximations that exactly satisfy the solenoidal requirement. A Galerkin basis is constructed and, starting with the solution of the associated Stokes problem, Newtonʹs method can be used to generate an approximate solution. A unique solution exists if the viscosity is sufficiently large; here an a posteriori computation is described that tests if uniqueness is likely to hold and gives an error estimate showing closeness of the approximation. Tests of the algorithm applied to problems in polygonal domains are described where approximations are 10th degree polynomials.