Dominik Schötzau

Superconvergence of the Local Discontinuous Galerkin Method for Elliptic Problems on Cartesian Grids

In this paper, we present a superconvergence result for the local discontinuous Galerkin (LDG) method for a model elliptic problem on Cartesian grids. We identify a special numerical flux for which the L2-norm of the gradient and the L2-norm of the potential are of orders k+1/2 and k+1, respectively, when tensor product polynomials of degree at most k are used; for arbitrary meshes, this special LDG method gives only the orders of convergence of k and k+1/2, respectively. We present a series of numerical examples which establish the sharpness of our theoretical results.

Local Discontinuous Galerkin Methods for the Stokes System

In this paper, we introduce and analyze local discontinuous Galerkin methods for the Stokes system. For a class of shape regular meshes with hanging nodes we derive a priori estimates for the L2-norm of the errors in the velocities and the pressure. We show that optimal-order estimates are obtained when polynomials of degree k are used for each component of the velocity and polynomials of degree k-1 for the pressure, for any

A locally conservative LDG method for the incompressible Navier-Stokes equations

k\ge1

DISCONTINUOUS GALERKIN FINITE ELEMENT METHOD FOR THE WAVE EQUATION

. We also consider the case in which all the unknowns are approximated with polynomials of degree k and show that, although the orders of convergence remain the same, the method is more efficient. Numerical experiments verifying these facts are displayed.

Optimal a priori error estimates for the hp -version of the local discontinuous Galerkin method for convection-diffusion problems

In this paper a new local discontinuous Galerkin method for the incompressible stationary Navier-Stokes equations is proposed and analyzed. Four important features render this method unique: its stability, its local conservativity, its high-order accuracy, and the exact satisfaction of the incompressibility constraint. Although the method uses completely discontinuous approximations, a globally divergence-free approximate velocity in H(div; Ω) is obtained by simple, element-by-element post-processing. Optimal error estimates are proven and an iterative procedure used to compute the approximate solution is shown to converge. This procedure is nothing but a discrete version of the classical fixed point iteration used to obtain existence and uniqueness of solutions to the incompressible Navier-Stokes equations by solving a sequence of Oseen problems. Numerical results are shown which verify the theoretical rates of convergence. They also confirm the independence of the number of fixed point iterations with respect to the discretization parameters. Finally, they show that the method works well for a wide range of Reynolds numbers.

ENERGY NORM A POSTERIORI ERROR ESTIMATION OF hp-ADAPTIVE DISCONTINUOUS GALERKIN METHODS FOR ELLIPTIC PROBLEMS

The symmetric interior penalty discontinuous Galerkin finite element method is presented for the numerical discretization of the second‐order wave equation. The resulting stiffness matrix is symmetric positive definite, and the mass matrix is essentially diagonal; hence, the method is inherently parallel and leads to fully explicit time integration when coupled with an explicit time‐ stepping scheme. Optimal a priori error bounds are derived in the energy norm and the

Mixed hp -DGFEM for Incompressible Flows

L^2

Time Discretization of Parabolic Problems by the HP-Version of the Discontinuous Galerkin Finite Element Method

‐norm for the semidiscrete formulation. In particular, the error in the energy norm is shown to converge with the optimal order

Interior penalty method for the indefinite time-harmonic Maxwell equations

{\cal O}(h^{\min\{s,\ell\}})

An hp -Analysis of the Local Discontinuous Galerkin Method for Diffusion Problems

with respect to the mesh size h, the polynomial degree