Optimal A Priori Error Estimates For The Hp-Version Of The Local Discontinuous Galerkin Method For Time-Dependent Convection-Diffusion Systems

Convection-diffusion equation can be used to solve the energy development, water conservancy engineering, fluid dynamics and electronic science, etc. And the convergence of the numerical method can indicate the quality of a numerical method. The convergence analysis of convection-diffusion problem has concerns of researchers.In this paper, we consider a simple nonlinear convection-diffusion problem as a model in a one-dimensional space domain; study the convergence properties for nonlinear convection-diffusion problem with Local Discontinuous Galerkin (LDG). We analysis error estimates of the problem, and obtain upper bounds for the energy norm of the error which are explicit in the mesh-size h and the polynomial degree p.This paper divides into 4 chapters.Chapter 1 simply introduces the background, present research situation and main results.In Chapter 2, we first give some definitions and lemmas can be used further, and then state the LDG method.In Chapter 3, we discrete the space using the idea of the discontinuous finite element, apply the LDG method to the model, and get the LDG scheme; we gain and show the theorem of stability by choosing a special numerical flux.In Chapter 4, by introducing an appropriate convection term numerical flux, projection operator we get the projection of error estimate, by hp-appropriate estimate lemma we finally get the error estimate expression on h,p under the energy norm.Finally, this paper also gives the corresponding numerical experiments results to verify the error estimate theorem, by analyzing the results comparatively; we find that the numerical results are consistent with theoretical results.