The Error Estimates Of Discontinuous Galerkin Method For Higher Order Partial Differential Equations

Due to its high convergence accuracy and numerical resolution,the discontinuous Galerkin finite element method has been widely used in the fields of fluid mechanics,environment and financial engineering in recent years.With the increasing complexity of the research on the real situation,in order to meet the real needs,further research on the general numerical circulation is needed.This thesis focuses on the stability and error estimates of the semi-discrete and fully discrete schemes of the local discontinuous Galerkin finite element method for high-order partial differential equations based on the upwind-biased numerical flux.First,for the fourth-order linear equation,using the generalized Gauss-Radau projection,the stability analysis and error estimation of the local discontinuous Galerkin finite element method based on the upwind-biased numerical fluxes are given.At the end of this chapter,the stability of the method and the k(10)1 order of convergence are verified through numerical experiments.Then,for the linear Kd V-Burgers equation,the stability analysis and error estimation of the discontinuous Galerkin finite element method using the upwind-biased numerical flux are given.Theoretical analysis of the convergence order reaches the k(10)1/2 order,and the numerical experiment verifies that the convergence order can achieve the convergence accuracy.Finally,the heat conduction equation is studied,and the stability of the equation's third-order explicit TVD Runge-Kutta local discontinuous Galerkin method is given,in which the upwind-biased numerical flux is selected for the numerical flux.In the case that the solution of the equation is sufficiently smooth,through the method of finite element analysis,it is strictly proved theoretically that the algorithm is L~2stable for any non-uniform regular grid and k subsection polynomial discontinuous finite element space.