Stability,Error Estimates And Superconvergence Analysis For Discontinuous Galerkin Method

In this thesis,the superconvergence properties of the semi-discrete discontinuous Galerkin(DG)method and the stability and error estimates of the fully-discrete DG scheme are studied,and the work contains three parts.The first part is that we perform the superconvergence analysis of the semi-discrete local discontinuous Galerkin(LDG)method for one-dimensional linear Schrodinger equation.The core idea of this work is to construct a special interpolation function by using the corresponding energy equation of the LDG scheme,and prove that the nu-merical solution is superclose to the interpolation function in the L2-norm.The order of superconvergence is 2k+1,when the polynomials of degree at most k are used.Even though the Schrodinger equation involves only second order spatial derivative,it is ac-tually a wave equation because of the coefficient i.Moreover,the finite element space of the LDG scheme is a complex valued function space.Compared with the parabolic case,the superconvergence analysis of the LDG method for Schrodinger equation is more difficult and complicated.By constructing special correction functions and suit-able initial discretization,we rigorously prove that the errors between LDG solutions and interpolation functions are superconvergent with a rate of(2k + 1)-th order,where the interpolation functions are defined in terms of correction functions and Gauss-Radau projections.With the help of this superconvergence result,we further obtain a(2k +1)-th order convergence rate for the domain average,cell averages and the pointwise error of numerical fluxes at nodes.In addition,we also establish that both the function value and the derivative approximations are superconvergent with a rate of(k+2)-th order at the Radau points.Numerical examples are shown to verify our theoretical results.The second part is that we study the stability and error estimates of a special fully-discrete scheme for the convection-diffusion equations,which consists of the semi-implicit spectral deferred correction(SDC)scheme in time and the LDG method in space.Based on the Picard integral equation,the SDC method is a class of time dis-cretization schemes and is driven iteratively by either the explicit or implicit Euler method.An advantage of the SDC method is that we can construct it easily for any order of accuracy.For the semi-implicit SDC scheme,the intermediate functions gen-erated by iterations and the left-most endpoint involved in the integral for the implicit part increase the difficulty of theoretical analysis for the fully-discrete scheme.To be more precise,the test functions are more complex and energy equations are more diffi-cult to construct,compared with semi-implicit Runge-Kutta schemes.By choosing test functions associated with the left endpoint and proper linear combinations of different layer functions,we prove that both the second and third order semi-implicit SDC time discretization coupled with LDG scheme are stable provided the time step is bounded above by some constant,which depends on diffusion and convection coefficients,but is independent of the mesh size.Based on the stability analysis,we obtain optimal error estimates in time and space for the corresponding fully-discrete schemes.Numerical examples are presented to illustrate our theoretical results.The last part is that we study the stability and error estimates of a fully-discrete scheme for linear conservation law on a moving grid.The grid moving methodology discussed here belongs to the class of arbitrary Lagrangian-Eulerian(ALE)methods,and we use the DG scheme to discretize spatial variables.Thus the spatial discretization is called a ALE-DG scheme.We use the explicit total variation diminishing Runge-Kutta schemes up to third order accuracy for time discretization and analyze the stability and error estimates of the corresponding fully-discrete schemes.The dependence of finite element space on time increases the difficulty of analysis.The scaling arguments and standard energy analysis are the key techniques used in our work.Under suitable CFL conditions and with Lax-Friedrichs numerical flux,we present a rigorous proof to obtain the stability for three fully-discrete schemes.Moreover,we derive the quasi-optimal error estimates in space and optimal convergence rates in time.