The construction of a null basis for a 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 the standard 9-node velocity and 4-node pressure DFFEM, a basis for the divergence-free subspace is constructed such that each basis function has nonzero support on at most 4 contiguous elements. Given this basis, discretely divergence free macro elements can be constructed and used in the implementation of the DFFEM.