Abstract. We provide a framework for the analysis of a large class of discontinuous methods for second-order elliptic problems. It allows for the understanding and comparison of most of the discontinuous Galerkin methods that have been proposed over the past three decades for the numerical treatment of elliptic problems.

A high-order discontinuous Galerkin finite element discretization and p-multigrid solution procedure for the compressible Navier-Stokes equations are presented. The discretization has an element-compact stencil such that only elements sharing a face are coupled, regardless of the solution space. This limited coupling maximizes the effectiveness of the p-multigrid solver, which relies on an element-line Jacobi smoother. The element-line Jacobi smoother solves implicitly on lines of elements formed based on the coupling between elements in a p = 0 discretization of the scalar transport equation. Fourier analysis of 2-D scalar convection-diffusion shows that the element-line Jacobi smoother as well as the simpler element Jacobi smoother are stable independent of p and flow condition. Mesh refinement studies for simple problems with analytic solutions demonstrate that the discretization achieves optimal order of accuracy of O(h p+1). A subsonic, airfoil test case shows that the multigrid convergence rate is independent of p but weakly dependent on h. Finally, higher-order is shown to outperform grid refinement in terms of the time required to reach a desired accuracy level.

In this paper, we introduce and analyze local discontinuous Galerkin methods for the Stokes system. For arbitrary meshes with hanging nodes and elements of various shapes 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 k >= 1. 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.

We present an algorithm to construct meshes suitable for space-time discontinuous Galerkin finite-element methods. Our method generalizes and improves the 'Tent Pitcher' algorithm of /lngSr and Sheffer. Given an arbitrary simplicially meshed domain X of any dimension and a time interval [0, T], our algorithm builds a simplicial mesh of the space-time domain X x [0, T], in constant time per element. Our algorithm avoids the limitations of previous methods by carefully adapting the durations of space-time elements to the local quality and feature size of the underlying space mesh.

Abstract. We study the convergence properties of the hp-version of the local discontinuous Galerkin finite element method for convection-diffusion problems; we consider a model problem in a one-dimensional space domain. We allow arbitrary meshes and polynomial degree distributions and obtain upper bounds for the energy norm of the error which are explicit in the mesh-width h, in the polynomial degree p, and in the regularity of the exact solution. We identify a special numerical flux for which the estimates are optimal in both h and p. The theoretical results are confirmed in a series of numerical examples. 1.

In this paper we propose a new general framework for the construction and the analysis of Discontinuous Galerkin (DG) methods which reveals a basic mechanism, responsible for certain distinctive stability properties of DG methods. We show that this mechanism is common to apparently unrelated stabilizations, including jump penalty, upwinding, and Hughes–Franca type residual-based stabilizations.

Abstract We present a non-diffusive and contact discontinuity capturing scheme for linear advection and compressible Euler system. In the case of advection, this scheme is equivalent to the Ultra-Bee limiter of [20], [24]. We prove for the Ultra-Bee scheme a property of exact advection for a large set of piecewise constant functions. We prove that the numerical error is uniformly bounded in time for such prepared (i.e. piecewise constant) initial data, and state a conjecture of non-diffusion at infinite time based on some local over-compressivity of the scheme for general initial data. We generalize the scheme to compressible gas dynamics and present some numerical results.

A Newton-Krylov method is developed for the solution of the steady compressible Navier-Stokes equations using a Discontinuous Galerkin (DG) discretization on unstructured meshes. An element Line-Jacobi preconditioner is presented which solves a block tridiagonal system along lines of maximum coupling in the flow. An incomplete block-LU factorization (Block-ILU(0)) is also presented as a preconditioner, where the factorization is performed using a reordering of elements based upon the lines of maximum coupling used for the element Line-Jacobi preconditioner. This reordering is shown to be far superior to standard reordering techniques (Nested Dissection, One-way Dissection, Quotient Minimum Degree, Reverse Cuthill-Mckee) especially for viscous test cases. The Block-ILU(0) factorization is performed in-place and a novel algorithm is presented for the application of the linearization which reduces both the memory and CPU time over the traditional dual matrix storage format. A linear p-multigrid algorithm using element Line-Jacobi, and Block-ILU(0) smoothing is presented as a preconditioner to GMRES. The coarse level Jacobians are obtained using a

this paper is to study the eects of the discontinuities in the uxes of the discontinuous Galerkin approximations of elliptic problems. Due to their exibility, discontinuous Galerkin (DG) methods have been popular among the nite element community and they have been applied to a wide range of computational uid problems. Since the rst DG method introduced in [16] the methods have been developed for hyperbolic problems, see [7] for an overview, and for elliptic problems in [21, 15, 8, 17, 18, 11]. A uni ed analysis for many DG methods has been given recently in [4]

A basic target algorithm for approximating unsteady hyperbolic conservation laws uses a finite volume formulation in three steps: recovery or reconstruction of a more accurate approximation from a set of cell averages; solution of the conservation law to obtain interface uxes averaged over a time step; and computation of new cell averages at the new time level. In this paper the target is achieved by bringing together ideas from Brenier's transport collapse operator -- using Lin et al's Riemann-Stieltjes interpretation, van Leer's MUSCL algorithm, Colella and Woodward's PPM algorithm and Goodman and LeVeque's flux approximation. First, second and third order accurate algorithms are developed for non-uniform one-dimensional grids and unstructured triangular meshes. The MUSCL-type scheme in one dimension is proved to be TV-stable right up to the natural CFL limit, that characteristics cross no more than one cell in one time step, and under the least restrictive necessary TVD condition on th...