Research On WG Finite Element Method For Convective Dominant Fluid Problem

Convection-diffusion equations and Navier-Stokes equations are the important models to describe the motion of viscosity fluid.They reflect the basic mechanical laws of viscosity fluid,so they play a vital role in the fluid dynamics.Then,convection-diffusion equations and Navier-Stokes equations are essential mathematical models in computational fluid dynamics,and their numerical methods are still growing.The aim of this thesis is to systematically study the weak Galerkin(WG)methods for the time-dependent convection-diffusion equations and time-dependent incompressible Navier-Stokes equations.In particular,the research is focused on developing robust and efficient WG methods and their error analysis on polygonal meshes for time-dependent linear/nonlinear convection-dominated diffusion equations and time-dependent incompressible Navier-Stokes equations.In Chapter 1,we give brief introduction of some preliminaries and notations used in finite element methods.Taking Poisson equation for example,weak Galerkin method and its error analysis for second order elliptic partial differential equations is presented.In Chapter 2,we present a robust and efficient weak Galerkin finite element method for time-dependent linear convection-dominated diffusion equations.Convection-diffusion equation includes a diffusion term with elliptic property and a convection term with hyperbolic property.In the case of convection-dominated diffusion equation,convection term is in a leading position,so its numerical discretization has an important impact on the robustness and non-physical oscillation-free numerical solution of the whole numerical scheme.To properly handle the convection term,weak directional derivative is introduced for a class of discontinuous functions defined on a polygonal mesh,which is used for the discretization of the convection term.And a stabilizer term with the advantage of upwinding characteristic is also designed for the convection term.These together with weak Galerkin finite element scheme of diffusion term yields the spatial semi-discrete weak Galerkin finite element discretization of time-dependent linear convection-dominated diffusion equations.Two fully discrete weak Galerkin methods are obtained by adopting implicit Euler scheme and Crank-Nicolson scheme respectively for time discretization.In Chapter 3,we generalize the weak Galerkin method and its stability and error estimate results for linear convection-diffusion equations presented in Chapter 2 to the nonlinear case.Compared with the linear convection-dominated diffusion equation,the numerical approximation of nonlinear convection-dominated diffusion is more difficult and challenge.The difficulty lies in the numerical approximation and analysis of the nonlinear convection term.First of all,the weak directional derivative is derived for the nonlinear convection term.And then a linearized stabilizer term with the property of upwinding characteristic is also designed for the nonlinear convection term.These two building blocks are used for the spatial semi-discrete weak Galerkin finite element methods for time-dependent nonlinear convection-dominated diffusion equations.A fully discrete weak Galerkin scheme is obtained by using implicit Euler scheme for the discretization of time derivative.The stability and error analysis of the semi-discrete weak Galerkin method and the fully discrete weak Galerkin method for time-dependent nonlinear convection-dominated diffusion equations are established on polygonal meshes.In Chapter 4,we further generalize the weak Galerkin method and error analysis techniques presented in Chapter 3 to the incompressible Navier-Stokes equations which have more wide applications in realistic world.The Navier-Stokes equation can be regarded as a system of nonlinear convection-diffusion equations.And it is also can be considered as a special structure vector valued nonlinear convection-diffusion equation.Weak gradient and weak directional derivative are introduced for vector-valued discontinuous function.Based on the variational formulation of the Navier-Stokes equation,a spatial semi-discrete weak Galerkin finite element scheme is derived for time-dependent incompressible Navier-Stokes equations.An implicit Euler scheme is used for the temporal discretization.The stability and error estimates are proved for both semi-discrete and fully discrete weak Galerkin finite element scheme of time-dependent incompressible Navier-Stokes equations.In Chapter 5,some numerical results are presented to confirm the valid of the theory and the accuracy of the numerical schemes analyzed in Chapter 2.The matrix form of the weak Galerkin finite element schemes is derived and then is used to test a series of numerical experiments.Numerical results validate the convergence theory developed in Chapter 2 for the weak Galerkin finite element method of time-dependent linear convection-dominated diffusion equations.