Preconditioned conjugate gradients for solving the transient Boussinesq equations in three-dimensional geometries

In this paper we present a new version of the ‘modified finite element method’ (MFEM) presented by Gresho, Chan, Lee and Upson.¹ The main modification of the original algorithm is the introduction of a cost-effective and memory-saving iterative solver for the discretized Poisson equation for the pressure. The vectorization of the preconditioner has been especially considered. For low Prandtl number problems we also split the advection-diffusion operator of the energy equation into explicit and implicit parts. In that sense the present approach is related to the recent implicitization of the diffusive terms introduced by Gresho and Chan² and by Gresho.³ The algorithm is applied to the study of buoyancy-driven flow oscillations occuring in a horizontal crucible of molten metal under the action of a horizontal temperature gradient.