Finite Element Approximation of Rotating MHD-Boussinesq Flows with Nonlinear Power-Law Viscosity

Abstract

We propose and analyse a fully discrete finite element approximation of the three-dimensional incompressible rotating MHD-Boussinesq system with nonlinear power-law ($p$-Laplace) viscosity. Conforming elements are used for the velocity and temperature, $\mathbf H(\operatorname{curl})$-conforming N\'ed\'elec elements for the magnetic induction, and a semi-implicit backward Euler scheme in time. The discretisation preserves the principal energy structure: the Lorentz and induction terms cancel exactly, while the Coriolis force remains energy neutral. This yields well-posedness and mesh-independent energy stability. The nonlinear power-law diffusion creates two main analytical difficulties. Since a uniform time-derivative estimate for the discrete velocity is unavailable, strong compactness is obtained through a time-translate argument and Simon's compactness criterion. The weak limit of the nonlinear diffusion is then identified by a Minty argument based on a discrete energy $\limsup$ inequality. Compactness and passage to the limit hold throughout the shear-thickening regime, while identification of a genuine weak solution is obtained in the Newtonian case and, in three dimensions, above an explicit threshold dictated by the Lorentz force rather than by convection. We further establish a priori error estimates without any mesh-time coupling condition. Three-dimensional numerical experiments confirm the stability and convergence behaviour predicted by the analysis.