Adjoint consistency – in addition to consistency – is the key requirement for discontinuous Galerkin discretisations to be of optimal order in L as well as measured in terms of target functionals. We provide a general framework for analysing adjoint consistency and introduce consistent modifications of target functionals. This framework is then used to derive an adjoint consistent discontinuous...

The DG algorithm is a powerful method for solving pdes, especially for evolution equations in conservation form. Since the algorithm involves integration over volume elements, it is not immediately obvious that it will generalize easily to arbitrary time-dependent curved spacetimes. We show how to formulate the algorithm in such spacetimes for applications in relativistic astrophysics. We also ...

Finite element methods for acoustic wave propagation problems at higher frequency result in very large matrices due to the need to resolve the wave. This problem is made worse by discontinuous Galerkin methods that typically have more degrees of freedom than similar conforming methods. However hybridizable discontinuous Galerkin methods offer an attractive alternative because degrees of freedom...

Abstract. We study the numerical solution for Volerra integro-differential equations with smooth and non-smooth kernels. We use an h-version discontinuous Galerkin (DG) method and derive nodal error bounds that are explicit in the parameters of interest. In the case of non-smooth kernel, it is justified that the start-up singularities can be resolved at superconvergence rates by using non-unifo...

Multiscale discontinuous Galerkin (DG) methods are established to solve flow and transport problems in porous media. The underlying idea is to construct local DG basis functions at the coarse scale that capture the local properties of the differential operator at the fine scale, and then to solve the DG formulation using the newly constructed local basis functions instead of conventional polyno...

In this paper, we develop functional a posteriori error estimates for DG approximations of elliptic boundary-value problems. These estimates are based on a certain projection of DG approximations to the respective energy space and functional a posteriori estimates for conforming approximations (see [30, 31]). On these grounds we derive two-sided guaranteed and computable bounds for the errors i...

Article history: Received 24 September 2014 Received in revised form 13 March 2015 Accepted 17 March 2015 Available online 24 March 2015

We propose a way of extending the discontinuous Galerkin method from pure hyperbolic equations to convection-dominated equations with an 0(h) diffusion term. The resulting method is explicit and can be applied with polynomials of degree n > 1 . The extended method satisfies the same 0(hn+ll2) error estimate previously established for the discontinuous Galerkin method as applied to hyperbolic pr...

In this paper, we develop and analyze the Runge-Kutta discontinuous Galerkin (RKDG) method to solve weakly coupled hyperbolic multi-domain problems. Such problems involve transfer type boundary conditions with discontinuous fluxes between different domains, calling for special techniques to prove stability of the RKDG methods. We prove both stability and error estimates for our RKDG methods on ...

The aim of this paper is to present and analyze a class of hpversion discontinuous Galerkin (DG) discretizations for the numerical approximation of linear elliptic problems. This class includes a number of well-known DG formulations. We will show that the methods are stable provided that the stability parameters are suitably chosen. Furthermore, on (possibly irregular) quadrilateral meshes, we ...