A Posteriori Error Estimation Of Finite Element Solutions For Burgers Equation And Its Application

As one of the most basic partial differential equation in the field of fluid mechanics,the Burgers equation is widely used in many fields.Although it is a nonlinear partial differential equation,we can find its exact solution in some cases.The solution of the Burgers equation may be oscillated in some place,which makes it very hard to obtain its numerical solution.Therefore,the investigation of the numerical method for Burgers equation is very important both in theory and application.A posterior error estimation based adaptive finite element method is very effective in solving the problem with local singularity.This paper has five chapters.The background of the adaptive finite element,the background and significance of the Burgers equation and the current state of the Burgers equation are given in the introduction.The second chapter provides the most needed inequality,theorem and the related knowledge of the Sobolev space and the mesh management strategy.In the third chapter and the fourth chapter,we deal with the one-dimensional and two-dimensional Burgers problem with Dirichlet boundary condition based on the Cole-hopf transformation.Then,the least squares finite element method is used to discrete the heat equation both in time and space.The semi-discrete and fully discrete schemes form are obtained.The a posteriori error estimator is constructed.The numerical results verify the validity of the theoretical analysis and the feasibility of the numerical method we proposed.One dimension we select a larger Reynolds number,compared with the error in the uniform grid.The results show that the number of adaptive grids is smaller and the computational efficiency is improved in the case of similar errors.In the case of two-dimensional cases,adaptive and uniform meshing is performed in the regions under different time nodes,The numerical simulation results show that the theory of this paper is correct and the numerical method of construction is feasible.Chapter 5 makes a summary and some prospect of the study.