A full discretisation of the time-dependent Boussinesq (buoyancy) model with nonlinear viscosity

In this article, we study the time dependent Boussinesq (buoyancy) model with nonlinear viscosity depending on the temperature. We propose and analyze first and second order numerical schemes based on finite element methods. An optimal a priori error estimate is then derived for each numerical scheme. Numerical experiments are presented that confirm the theoretical accuracy of the discretization.