Stability Analysis And Negative-Order Norm Estimates Of Discontinuous Galerkin Methods For Convection-Diffusion Equations

Discontinuous Galerkin(DG)methods are a kind of numerical method of high-resolution partial differential equations,there is a wide range of applications in engineering calculations due to its advantages such as the ability to achieve arbitrary order accuracy,h-p adaptability,the ability to handle complex calculation areas and provable L~2-norm stability.So it is of great significance to the theoretical study of the intermittent Galerkin method.This paper mainly studies the stability analysis and negative-order norm estimation of the discontinuous Galerkin method for convection-diffusion equation.The paper first introduces the basic properties of the discontinuous Galerkin finite element space,and proves the L~2-norm stability of the second-order explicit TVD Runge-Kutta discontinuous Galerkin method under the divided difference for the heat conduction equation.Then for the nonlinear convection-diffusion equation,Taylor expansion linearization is used to deal with the nonlinear numerical flux,and it is proved that when the upwind-type numerical flux is used,the ?-th order divided difference of the DG error could achieves the accuracy of order k+3/2-?/2 in L~2-norm.Furthermore,the dual proof method is used to prove that the divided difference of the DG error can reach the(2k+3/2-?/2)-th order superconvergence accuracy in negative-order norm.It is proved that the post-processing theory is applied to the nonlinear convection-diffusion equation,and the post-processing solution can obtain at least(3k/2+1)-th order superconvergence accuracy.Numerical experiments verify the correctness of the theoretical results.Finally,the problem of the negative-order norm estimate for the variable coefficient convection-diffusion equation is studied.Similar to the proof of the nonlinear equation,it is proved that the divided difference of the DG error can reach the(k+1)-th order accuracy in the L~2-norm,and then the(2k+1)-th order can be obtained in the negative-order norm.It is proved that the superconvergence accuracy of the post-processing numerical solution is also(2k+1)-th order in final,and the theoretical results are verified by numerical experiments.