The present numerical study deals with the stabilized finite element solution of the steady natural convection flow in terms of primitive variables. The domain is discretized into a set of regular linear triangular elements for all unknowns containing a single subgrid node, which has a crucial effect in the stabilization with a cheap computational cost. The proposed stabilized finite element method enables obtainment of a stable solution and avoids oscillations, especially in the pressure. The algorithm is presented for different benchmark problems considered in different geometries and the solutions are obtained for high values of Rayleigh number (Ra) (10³ ≤ Ra ≤ 10⁶). Furthermore, the solutions show the well-known characteristics of the natural convection flow and agree well with the results in the literature.

This study deals with the stabilized finite element solution of the steady, natural convection flow in an enclosure under a magnetic field applied perpendicular to the sinusoidal corrugated vertical walls of the enclosure, in terms of primitive variables. Several vertical sinusoidal functions are selected for the comparison. A stabilized FEM scheme called SSM is proposed in order to obtain stable solution for the high values of problem parameters with a cheap computational cost. Proposed numerical scheme with linear elements is tested with the other well-known stabilization method called SUPG and also with the higher order element combinations ( $Q_2-Q_1$). Obtained solutions show the better accuracy of SSM compared to SUPG in terms of the $L_2$ norm and also show the well-known characteristics of the natural convection MHD flow. The decrease in the convection heat transfer with an increase in the size of the heat source placed in bottom wall is observed.