Unconditionally stable Gauge-Uzawa finite element schemes for incompressible natural convection problems with variable density

We construct in this paper two Gauge-Uzawa schemes, one in conserved form and the other in convective form, for solving natural convection problems with variable density, and prove that the first-order versions of both schemes are unconditionally stable. We also show that a full discretized version of the conserved scheme with finite elements is also unconditionally stable. These schemes lead to a sequence of decoupled elliptic equations to solve at each step, hence, they are very efficient and easy to implement. We present several numerical tests to validate the analysis and demonstrate the effectiveness of these schemes for simulating natural convection problems with large density differences.