The Finite Element Numerical Methods For Two-Dimensional Burgers Equation

The finite element method is a very important and effective numerical method for solving the partial differential equations. Finite element method was first proposed by R. Courant in 1943. The mathematical ideas are variational principle or equation allowance and weight function of orthogonalization principle. With the rapid development of computer technology, finite element method has gotten a very rapid development and applied to many fields, such as aviation technology, manufacturing, construction and medicine etc.Burgers equation is one of models of nonlinear convection diffusion phenonmena. However, generally speaking, this well-posed problem can lead to a shock wave with large Reynolds number, which makes it difficult to solve Burgers equation. Therefore, there are significant in academic meaning and applied value to research the numerical methods for this equation.This thesis mainly studies the numerical methods for nonlinear Burgers equation and corresponding stability analysis. Firstly we introduce the background and significance of the Burgers equation and point out the difficulties of simulating Burgers equation. Then, the thesis simply introduces the development situation of three methods that are used in this article.We discuss the operator-splitting Galerkin finite element method to solve nonlinear Burgers equation, which is split into pure convection equation and diffusion equation to avoid computing difficulty of two different physical processes. Firstly, the operator splitting discontinuous finite element method of Burgers equation is discussed, and the solution of the existence and uniqueness is obtained. Secondly, we give a semi-implicit operator splitting numerical method to solve the equation. The single-step scheme and its multistep variant are derived for the Burgers equation. The numerical experiments are carried out to check the accuracy, stability, performance of this method.Based on Hopf-Cole that can transform the nonlinear Burgers equation to linear diffusion equation with Neumann boundary conditions, the LDG finite element method for nonlinear Burgers equation is introduced. Then the formulation, stability analysis, entropy inequality are presented.Using the parabolic equation of punishment in the form of discontinuous finite element method, we establish the punishment form of discontinuous finite element method for Burgers equation. Then we give the estimate error and the stability analysis of the method.