Abstract
A class of finite element schemes for nonstationary thermal convection problems with temperature-dependent coefficients are considered. Such variable coefficients make the diffusion and the buoyancy terms nonlinear, whose estimates are key points in our analysis. Because of an argument based on the energy method, optimal error estimates for the velocity and the temperature without any stability conditions have been established. A model of the practical melting glass convection is then computed by a finite element method based on such a scheme mathematically justified. Some numerical results are demonstrated to investigate differences of the energy efficiency among some data configurations.
