Computation of incompressible thermal flows using Hermite finite elements

Using Hermit basis functions with the finite element method offers a remarkably simple way to compute non-isothermal buoyancy-driven incompressible flow. The Hermite bases we use simplify the governing equation and strongly enforce the continuity equation. For this problem, we use a fourth-order C1 stream function defined on rectangles here, but other higher and lower-order Hermite elements on rectangles and triangles can easily be derived or modified from elements found in the plate-bending literature. Hermite elements are also used for the temperature and pressure. We conclude with results from application of the method to the square thermal cavity at moderate to high Rayleigh numbers.