A stabilized finite element method for the Rayleigh–Bénard equations with infinite Prandtl number in a spherical shell Original Research Article

A finite element scheme is developed and analyzed for a thermal convection problem of Boussinesq fluid with infinite Prandtl number in a spherical shell. This problem is a mathematical model of the Earthʹs mantle movement and has been a topic of interest for geophysicists. It is described by the Rayleigh–Bénard equations with infinite Prandtl number, that is, a system of the Stokes equations and the convection–diffusion equation coupled with the buoyancy and the convection terms. A stabilized finite element scheme with P1/P1/P1 element is presented, and an error estimate is established. The obtained theoretical convergence order is also recognized by a numerical result. Another numerical result is shown as an example of the Earthʹs mantle movement simulation.