Two Finite Element Methods For Natural Convection Problem With Variable Density

Incompressible natural convection problems with variable density are widely used in many fields such as thermodynamics and geophysics.The equations have some numerical difficulties such as complex coupling,nonlinear and hyperbolic prop-erty.Therefore,it is very important to study the efficient numerical algorithms.We will consider the incompressible natural convection equation with variable density and construct high efficient finite methods to overcome these difficulties.The high Rayleigh number problem has always been one of the difficulties in the incompressible natural convection equation with variable density.Besides,the equation have nonlinear terms.If we use iterative methods to deal with it,it takes a long time.So,we introduces a novel characteristic variational multiscale finite element method which combines advantages of both characteristic and variational multiscale methods within a variational framework for solving the incompressible natural convection problem with variable density.The main novel ideas of this work are to overcome the stability issue due to the nonlinear inertial term and the hyper-bolic term for conventional finite element methods,and to deal with high Rayleigh number for the natural convection problem.We choose the conforming finite ele-ment pair?P₂,P₂,P₁,P₂?to approximate the density,the velocity,the pressure and the temperature field,respectively.Moreover,the stability analysis of the C-VMS method is given.Finally,ample numerical results are presented to demonstrate the efficacy and accuracy of the proposed method.The pressure projection finite element method is proposed for the strong cou-pling and nonlinear of the incompressible natural convection equation with variable density.The pressure projection method does not need iteration,so it can save time.The main novel ideas of this work are to overcome the stability issue due to the nonlinear inertial term and the hyperbolic term for conventional finite element methods.Moreover,the stability analysis of the first-order pressure-correction pro-jection finite element method is given.Finally,ample numerical results are presented to demonstrate the efficacy and accuracy of the proposed method.