Penalty decoupled iterative methods for the stationary natural convection equations with different Rayleigh numbers

Abstract

In this paper, we propose and analyze six combined schemes in the framework of penalty method to solve the natural convection (NC) equations numerically. The key role of the penalty technique is to manage the incompressibility constraint divu=0 effectively and improve the positivity of the resulting system. For definiteness and without loss of generality, we mainly explore a inf-sup stable finite element pair P1b-P1-P1b and a inf-sup unstable one P1-P0-P1. Three efficient iterative schemes are applied to deal with the nonlinear terms. Moreover, the results of the discrete stability and optimal convergence under the exactly different stability conditions are proved for the given methods. Numerical simulations and results for three benchmark problems are explored to illustrate the theoretical findings.