The derivation of minimal support basis functions for the discrete divergence operator

The divergence free finite element method (DFFEM) is a method to find an approximate solution of the Navier-Stokes equations in a divergence free space. That is, the continuity equation is satisfied a priori. DFFEM eliminates the pressure from the calculations and significantly reduces the dimension of the system to be solved at each time step. For a low order and nonconforming element, we construct a basis for the divergence free subspace such that basis functions have smallest nonzero support.