Error Estimate Of Gauge-Uzawa Methods For Incompressible Navier-Stokes Equation With Variable Density

The incompressible Navier-Stokes(N-S)equation is one of the most common physical models and is used in fluid mechanics to describe the motion of incompressible fluids.It is also used in many practical projects.The existence and smoothness of the solution have not been fully proved so far.Therefore,the numerical algorithm for studying the equation has practical theoretical and practical significance.In this paper,we mainly use the finite element algorithm to construct and analyze the incompressible N-S equation with variable density.The main contents include the following aspects:In chapter 1,we give the research background and current situation of the thesis,re-view some existing research results of the N-S equation,and give some relevant preliminary knowledge.In chapter 2,we review the Gauge-Uzawa method(GUM)for solving incompressible N-S equation with variable density.The positivity-preserving and bounded of the numerical density is proved by its stability,which establishes the basis for subsequent analysis.In chapter 3,the convergence of discrete velocity and pressure in the convection GUM scheme is analyzed respectively,and the convergence order is given.In chapter 4,we construct a new fully discrete finite element algorithm and analyze its positivity-preserving and stability.Finally,in the last chapter,some numerical experiments are presented to validate the proposed theoretical results.