We lay out a program for constructing discontinuous Petrov-Galerkin (DPG) schemes having test function spaces that are automatically computable to guarantee stability. Given a trial space, a DPG discretization using its optimal test space counterpart inherits stability from the well-posedness of the undiscretized problem. Although the question of stable test space choice had attracted the atten...

A new space-time discontinuous Galerkin (dG) method utilizing special Trefftz polynomial basis functions is proposed and fully analyzed for the scalar wave equation in a second order formulation. The dG method considered is motivated by the class of interior penalty dG methods, as well as by the classical work of Hughes and Hulbert [Comput. Methods Appl. Mech. Engrg., 66 (1988), pp. 339–363; Co...

This paper is an attempt in seeking a connection between the discontinuous Petrov– Galerkin method of Demkowicz and Gopalakrishnan [13,15] and the popular discontinuous Galerkin method. Starting from a discontinuous Petrov–Galerkin (DPG) method with zero enriched order we re-derive a large class of discontinuous Galerkin (DG) methods for first order hyperbolic and elliptic equations. The first ...

In this paper we derive a priori L∞(L2) and L2(L2) error estimates for a linear advection-reaction equation with inlet and outlet boundary conditions. The goal is to derive error estimates for the discontinuous Galerkin (DG) method that do not blow up exponentially with respect to time, unlike the usual case when Gronwall’s inequality is used. While this is possible in special cases, such as di...

The entropy solutions of the compressible Euler equations satisfy a minimum principle for the specific entropy [11]. First order schemes such as Godunov-type and Lax-Friedrichs schemes and the second order kinetic schemes [6] also satisfy a discrete minimum entropy principle. In this paper, we show an extension of the positivity-preserving high order schemes for the compressible Euler equations...

In this paper, we present a Hybridizable Discontinuous Galerkin (HDG) method for the solution of the compressible Euler and Navier-Stokes equations. The method is devised by using the discontinuous Galerkin approximation with a special choice of the numerical fluxes and weakly imposing the continuity of the normal component of the numerical fluxes across the element interfaces. This allows the ...

This report presents a mesh adaptation method for higher-order (p > 1) discontinuous Galerkin (DG) discretizations of the two-dimensional, compressible Navier-Stokes equations. The method uses a mesh of triangular elements that are not required to conform to the boundary. This triangular, cut-cell approach permits anisotropic adaptation without the difficulty of constructing meshes that conform...

We propose and analyze a new hybridizable discontinuous Galerkin (HDG) method for second-order elliptic problems. Our method is obtained by inserting the L-orthogonal projection onto the approximate space for a numerical trace into all facet integrals in the usual HDG formulation. The orders of convergence for all variables are optimal if we use polynomials of degree k + l, k + 1 and k, where k...

We propose and analyse a new finite-element method for convection–diffusion problems based on the combination of a mixed method for the elliptic and a discontinuous Galerkin (DG) method for the hyperbolic part of the problem. The two methods are made compatible via hybridization and the combination of both is appropriate for the solution of intermediate convection–diffusion problems. By constru...

In this paper, we focus on error estimates to smooth solutions of semi-discrete discontinuous Galerkin (DG) methods with quadrature rules for scalar conservation laws. The main techniques we use are energy estimate and Taylor expansion first introduced by Zhang and Shu in [24]. We show that, with P k (piecewise polynomials of degree k) finite elements in 1D problems, if the quadrature over elem...