Error Estimates And Superconvergence Analysis Of Discontinuous Galerkin Methods

In this thesis,we mainly study the error estimates of discontinuous Galerkin(DG)methods,including priori error estimates,negative norm estimates and superconver?gence analysis,and design the numerical schemes to solve the spatial high order time dependent equations.This thesis consists of the following three parts.Firstly,we consider the error estimate and post-processing of local discontinuous Galerkin method for Schrodinger equations.Post-processing technique is applied only at the final time T of the DG approximation,and solutions after post-processing is formed through convolution with a kernel function which is a linear combination of central B-splines.After post-processing we could improve the accuracy and smooth-ness.The error estimate of the solution after post-processing depends on priori error estimates and negative norm estimates of the error between numerical solutions and ex-act solutions.Therefore,we give the L2 error estimate and then by the dual equations to get the estimate in the negative norl.We prove that in L2 nor1 local discontinu-ous Galerkin(LDG)solutions have the optimal order k+1 and higher order 2k in the negative norm,where k is the highest order of polynomials.We also give numerical examples to confirm our theoretical findings,including linear,variable coefficient and nonlinear cases in one and two dimensions.Then we combine the advantages of the LDG method and the ultra-weak DG(UWDG)method,construct the ultra-weak local DG(UWLDG)method to solve the spacial high order equations.We rewrite the PDEs with higher order spatial deriva-tives into a lower order system,then apply the UWDG method to the system.We first consider the fourth order and fifth order nonlinear PDEs in one space dimension,and then extend our method to general high order problems and two space dimensions.The main advantage of our method over the LDG method is that we have introduced fewer auxiliary variables,thereby reducing memory and computational costs.The main ad-vantage of our method over the UWDG method is that no internal penalty terms are necessary in order to ensure stability for both even and odd order PDEs.We prove sta-bility of our method in the general nonlinear case and provide optimal error estimates for linear PDEs for the solution itself as well as for the auxiliary variables approximat-ing its derivatives.Then we consider the superconvergence of the UWLDG scheme for an one-dimensional linear fourth-order equation by constructing correction functions.We also apply UWLDG methods to solve a class of nonlinear fourth-order wave equa-tions,and prove the energy conserving property of our scheme and its optimal error estimates in the L2-norm for the solution itself as well as for the auxiliary variables ap-proximating the derivatives of the solution.Compatible high order energy conserving time integrators are also proposed.All theoretical findings are confirmed by numerical experiments.Finally,we consider the arbitrary Lagrange Euler discontinuous Galerkin(ALE-DG)method for one dimensional conservation laws.First,we consider the supercon-vergence analysis of the linear case.We start form the bilinear form of the ALE-DG scheme,and construct some correction functions to correct the error between the exact solutions and their projections.Based on these correction functions we could define a special interpolation function which is superclose to the numerical solution,and then we can prove the superconvergence results for cell averages,and some special points.Sec-ond,as another superconvergence result,we also study the negative norm estimate of the ALE-DG schemes for the nonlinear conservation laws.By the suitable dual equa-tion we prove 2k+1 accuracy in the negative norm.Recently,there has been many superconvergence studies for DG schemes,and for ALE-DG method,as an extension of the DG method,we also hope there are some superconvergence results.However,the ALE-DG method discussed here is a mesh moving method,this makes our work more complicated than for fixed grid,which we need to deal with the new challenges brought by time-dependent space and grid velocity field.Since the time derivative dose not commute with the space projections for the ALE-DG method,we need to introduce the material derivative in our analysis.With the help of the material derivative,we can prove the desired superconvergence results concisely and efficiently under suitable initial conditions.Therefore,the scaling argument and material derivative are the key techniques used in our work.