Error analysis for finite element methods for steady natural convection problems

We analyze the error in finite element methods in approximating, so-called, free or natural convection problems. We also include the effects of conducting solid walls in our analysis. Under a uniqueness condition on the Rayleigh and Prandtl numbers (which we derive), we give direct, quasioptimal error estimates for “div-stable” finite element spaces for the fluid variables and general conforming finite element spaces for the temperature. At larger Rayleigh numbers, we give analogous, asymptotic error estimates, basing this analysis upon local uniqueness properties of the true solution (u p T), which we also establish.