The present work consists of comparative evaluation of the finite-element and finite-difference methods for the solution of complex fluid flow and heat transfer problems. The test case considered here is the problem of natural convection in an annular cavity. Both closed and open ended cases are considered. It was found that the finite-difference analysis succeeds in capturing minute details of the solution at high Rayleigh number flows and the finite-element method fails to do so. However, the finite-element method enjoys the advantage of providing sufficiently accurate results even with coarse meshes. This results in significant savings in computational costs without any sacrifice in the accuracy of the solution. Hence, the finite-element method can serve as an optimum tool for most problems of practical interest.