We describe the application of a local discontinuous Galerkin method to the numerical solution of the three-dimensional shallow water equations. The shallow water equations are used to model surface water flows where the hydrostatic pressure assumption is valid. The authors recently developed a DGmethod for the depth-integrated shallow water equations. The method described here is an extension ...

We present in this paper a new a posteriori error estimator for the Baumann-Oden version of the Discontinuous Galerkin Method. The error estimator is based on the residual of the partial differential equation. In the case of the reaction-diffusion equation, the norm of the residual is shown to be equivalent to the error in some specific energy-type norms. We propose here a method to efficiently...

We develop a high-order positivity-preserving discontinuous Galerkin (DG) scheme for linear Vlasov-Boltzmann transport equations (Vlasov-BTE) under the action of quadratically confined electrostatic potentials. The solutions of such BTEs are positive probability distribution functions and it is very challenging to have a mass-conservative, high-order accurate scheme that preserves positivity of...

We prove local and pointwise error estimates for the local discontinuous Galerkin method applied to the Stokes problem in two and three dimensions. By using techniques originally developed by A. Schatz [Math. Comp., 67 (1998), 877-899] to prove pointwise estimates for the Laplace equation, we prove optimal weighted pointwise estimates for both the velocity and the pressure for domains with smoo...

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...