Ziqing Xie

Error analysis of a discontinuous Galerkin method for Maxwell equations in dispersive media

A discontinuous Galerkin method for the numerical approximation of time-dependent Maxwell equations in three different dispersive media is introduced. Both the L^2-stability and error estimate of the DG method are discussed in detail. We show that the proposed method has an accuracy of O(h^k^+^1^2) under the L^2-norm when polynomials of degree k in space are used. Furthermore, numerical experiments are provided to justify our theoretical analysis.

Superconvergence of Discontinuous Galerkin Methods for Convection-Diffusion Problems

Some discontinuous Galerkin methods for the linear convection-diffusion equation −εu″+bu′=f are studied. Based on superconvergence properties of numerical fluxes at element nodes established in some earlier works, e.g., Celiker and Cockburn in Math. Comput. 76(257), 67–96, 2007, we identify superconvergence points for the approximations of u or q=u′. Our results are twofold:1) For the minimal dissipation LDG method (we call it md-LDG in this paper) using polynomials of degree p, we prove that the leading terms of the discretization errors for u and q are proportional to the right Radau and left Radau polynomials of degree p+1, respectively. Consequently, the zeros of the right-Radau and left-Radau polynomials of degree p+1 are the superconvergence points of order p+2 for the discretization errors of the potential and of the gradient, respectively.2) For the consistent DG methods whose numerical fluxes at the mesh nodes converge at the rate of O(hp+1), we prove that the leading term of the discretization error for q is proportional to the Legendre polynomial of degree p. Consequently, the approximation of the gradient superconverges at the zeros of the Legendre polynomial of degree p at the rate of O(hp+1).Numerical experiments are presented to illustrate the theoretical findings.

Space-time discontinuous galerkin method for maxwell equations in dispersive media

Abstract In this paper, a unified model for time-dependent Maxwell equations in dispersive media is considered. The space-time DG method developed in [29] is applied to solve the underlying problem. Unconditional L 2 -stability and error estimate of order O (τ r+1 + h k+1/2 ) are obtained when polynomials of degree at most r and k are used for the temporal discretization and spatial discretization respectively. 2-D and 3-D numerical examples are given to validate the theoretical results. Moreover, numerical results show an ultra-convergence of order 2 r +1 in temporal variable t .

The Highest Superconvergence Analysis of ADG Method for Two Point Boundary Values Problem

In this paper, an averaging discontinuous Galerkin (ADG) method for two point boundary value problems is analyzed. We prove, for any even polynomial degree k, the numerical flux convergence at a rate of

EXISTENCE OF MULTIPLE SOLUTIONS FOR NONLINEAR ELLIPTIC EQUATIONS WITH MIXED BOUNDARY CONDITIONS

2k+2

Superconvergence of DG method for one-dimensional singularly perturbed problems

2k+2 for all mesh nodes (in particular, the numerical flux for

Space-time discontinuous Galerkin method for Maxwell's equations

k=0