The Error Estimates And Superconvergences Of Discontinuous Galerkin Finite Element Method

Discontinuous Galerkin methods are a class of finite element method with the finite element space as the piecewise polynomial space which allows completely discontinuity.This method shares the advantages of high-order accuracy,being easy to implement h-p adaptive and apply in complicated geometries.Therefore,the DG method is widely applied in engineering field and it is meaningful to investigate the theory of the numerical method.In this paper,we mainly study the error estimates and superconvergence of discontinuous Galerkin method for three types of time dependent partial differential equations.Those are linear hyperbolic equations with varieties coffiesents,nonlinear convection-diffusion equations and linear fifth-order partial differential equations.we demonstrate the optimal error estimate of discontinuous Galerkin method with the upwind numerical fluxes for linear hyperbolic equations with varieties coefficients by constructing the special projections and finite element analysis techniques.Particularly,in the numerical experiments at the end of this chapter,we find out the fact that the numerical solutions are closer to these special projections than the exact solutions,which implies the superconvergence property.These results provide a new direction of our further investigations.For the nonlinear convection-diffusion equations with sign-preserved derivatives of convection terms,we prove that the solutions of discontinuous Galerkin methods convergent to the Gauss-Radau projections of the exact solutions at a rate of order,which is half order higher than optimal order.Some results of numerical experiments indicate the superconvergence property actually holds true.Finally,we use a new process of choosing the test functions to obtain the linear type of optimal error estimate and the superconvergence of local discontinuous Galerkin method for a class of linear fifth-order partial differential equations.The linear type error estimates and superconvergence imply that the errors of discontinuous Galerkin methods will not grow much in a long time.But some numerical results indicate the order we obtained is not optimal.