I have been solving the problem of natural convection in a 3D cubical cavity heated from below and I have been using a Galerkin spectral method whose basis and trial functions are divergence free functions which also satisfy boundary conditions. These functions are related to beam functions, but as the beam functions are the eigenvectors of a regular Sturm-Liouville problem the coefficients of the expansions decays algebraically as n goes to infinity. So I would like to work with divergence free functions got as eigenvectors of a singular Sturm-Liouville problem, such as chebyshev polynomials. Of course it is not possible to work with plain chebyshev polynomials because they do not verify boundary conditions. Does anybody know if it is possible to use any combination of chebyshev polynomials which satisfy both boundary conditions and divergence free condition? I would appreciate any information related to 3D free divergence functions used together with spectral methods.

A. Gelfgat from technion, haifa, israel, has built divergence free 2D vector basis function, both in cartesian and polar coordinates using Chebyshev polynomials of the 1st and 2nd kind

his email is alexg@cmlp.technion.ac.il