Piecewise Solenoidal Vector Fields and the Stokes Problem

Nonconforming finite element approximations to solutions of the Stokes equations are constructed. Optimal rates of convergence are proved for the velocity and pressure approximations. For the pressure approximation, $C^0 $ piecewise polynomial functions are used. The class of vector fields used to approximate the velocity field have piecewise polynomial components, discontinuous across interelement boundaries. On each “triangle” these vector fields satisfy the incompressibility condition pointwise. It is shown that these piecewise solenoidal vector fields possess optimal approximation properties to smooth solenoidal vector fields on domains with curved boundaries.