Error Estimates For Nonlinear Equations Using Discontinous Galerkin Methods And Post Processing Technique

The priori error analysis and post-processing technique for the discontinuous Galerkin (DG) methods for nonlinear partial differential equations (PDEs) have been studied in the present thesis. Most of the work presented originates from the published and sub-mitted papers within the past five years of my PhD study.In the prior error analysis, two categories of4th order strongly nonlinear parabolic evolution equations are studied, namely the surface diffusion equation and the Willmore flow equation. It is considered as a continued work of the preceding numerical study on these two equations using local discontinuous Galerkin (LDG) methods by Xu and Shu. The idea of applying LDG method is to rewrite the4th order equation into first order system, which splits the equation into linear parts containing high order derivatives and nonlinear parts. The treatment of the nonlinear terms is vital towards the prior error estimates and the derivation is expressed in detail. The results reveal that LDG methods can obtain an optimal accuracy order of k+1in L2-norm with discretization on Cartesian meshes where a completely discontinuous piecewise polynomial space of degree k≥1is applied.The post-processing technique is a topic based on the superconvergence property of finite element methods aiming to achieve a solution with higher order of accuracy and better smoothness. It is carried out by means of convolving the finite element solu-tion with a local averaging kernel. The present work is the attempt to implement LDG post-processing technique into linear parabolic equation and nonlinear hyperbolic equa-tion and considered further extension of available implementations into linear elliptic equation and linear hyperbolic equation.The multi-dimensional linear convection-diffusion equation is chosen when apply-ing the post-processing technique into LDG method for the linear parabolic equation. Theoretical derivation proves that an accuracy order of2k+m can be achieved in the negative-order norm, where m depends upon the flux and takes on the values0,1/2, or1. The LDG solution of the intended equation is later post-processed using a proper designed kernel. The resultant accuracy order in L2-norm is demonstrated the same as that of the liner hyperbolic equation in the literature. Simulations further confirm the improvement of accuracy order from (?)(hk+1) to (?)(h2k+1) with the employment of current post-processing technique. Extension of DG post-processing technique into nonlinear problem is for the first time attempted. The scalar nonlinear hyperbolic conservation law equation is consid-ered. An analysis of the error in negative-order norm is obtained by using technical dual argument. The accuracy order in negative-order norm is2k+m higher than the k+m order in the L2-norm. In this way, the negative-order norm estimates for linear hyperbolic equations in[15] are generalized. In order to do the post-processing, the treat-ment of the DG error in negative-order norm is essential and involves more difficulties. Simulation tests are carried out to validate that current post-processing technique can produce higher order accuracy provided the solution is smooth enough.