Error Estimates On The Local Discontinuous Garlerkin Meth-ods

There are wide applications for the time-dependent convection-diffusion prob-lems, for which lots of numerical methods have been studied and developed. Local discontinuous Galerkin (named LDG for short) method is one of the present pop-ular numerical method, it has good numerical stability and high-order accuracy. The purpose of this dissertation is to consider the optimal error estimate for the LDG method when solving the one dimensional and two dimensional convection-diffusion problems. The main conclusion contains two parts. One is the general setting of the numerical flux. Namely, we will adopt the generalized alternating numerical flux. The other is the "double-optimal" local error estimate.This dissertation is composed of seven chapters. In the first chapter, we present a review of the development of LDG methods, mainly for the convection diffusion equations. In the last chapter, we give some concluding remarks and the on-going work. The remaining five chapters are the main body of this dissertation, which are organized as follows.In the second chapter, we shall consider the one dimensional linear convection diffusion problem with the sufficiently smooth solution in the global domain. Based on the generalized alternating numerical flux, we shall prove that the LDG method still has the optimal global error estimate in L2-norm. To do that, we will adopt a global projection, which is introduced recently and named as the generalized Gauss-Radau (GGR) projection in this paper. As a main development of this projection, we improve the regularity assumption on the given function, in order to achieve the optimal approximation property of GGR projection in the measure of L2-norm.In the third chapter, we shall extend the previous study to the LDG method of the multidimensional convection diffusion problems. For simplicity, the discon-tinuous finite element space is made up of the piecewise tensor product polyno-mials on the Cartesian meshes. When the generalized alternating numerical flux is used, we will prove that the LDG method still has the optimal error estimate in the L2-norm. The main tool is the two-dimensional GGR projection. However, the new difficulty will cause due to the increase of the dimensions. In this case, we have to set up the superconvergence property of the two dimensional GGR projection with respect to the DG spatial discretization operator. Compared with the original local Gauss-Radau projection, the proof’s line of the corresponding conclusion is different.Starting from the fourth chapter, we will discuss the local error estimate for the LDG method. At first, we consider the singularly perturbed problem with a boundary layer in the fourth chapter. Since the considered exact solution is assumed to be non-smooth in the global domain, namely, the regularity depends strongly on the diffusion coefficient, the previous global error estimate seems useless and we have to conduct the corresponding local analysis to show the good numerical behavior of the LDG methods. The purpose of this dissertation is to build up the "double-optimal" local error estimate for the LDG method, when the exact solution varies quickly with a huge gradient in a narrow region and forms a fixed boundary layer. In other words, the pollution region, where the numerical solution is not good enough because of the existence of the boundary layer, has the nearly optimal width, and the L2-norm error is still optimal far away from the pollution region. To complete the corresponding proof, we need to present the energy analysis with a special weight function. The key techniques have three issues. Firstly, we set up the weighted stability in L2-norm by the help of the local L2 projection. Secondly, we redefine the GGR projection to deal with the Dirichlet boundary condition. Thirdly, using the regularity assumption of the exact solution, we give the specific setting of the parameters including in the weight function.In the fifth chapter, we consider the fully-discrete LDG method for the one dimensional singularly perturbed problems, where the time is updated by the second order and/or the third order total variation diminishing explicit Runge-Kutta algorithms. The key point of the analysis is to control effectively the information coming from the time discretization. Due to the different stability mechanism of two time-discretizations, there are evidently difference of the local error estimate for the above two fully-discrete LDG method.In the sixth chapter, we shall extend the "double-optimal" local error esti-mate of the LDG methods to the two dimensional singularly perturbed problems, where the semi-discrete LDG method with the purely alternating numerical fluxes is considered. In this situation, it is crucial to set up the weighted superconver-gence property of two-dimensional GR projection.