We present guaranteed and computable both sided error bounds for the discontinuous Galerkin (DG) approximations of elliptic problems. These estimates are derived in the full DG-norm on purely functional grounds by the analysis of the respective differential problem, and thus, are applicable to any qualified DG approximation. Based on the triangle inequality, the underlying approach has the foll...

In this paper, we adapt a simple weighted essentially non-oscillatory (WENO) limiter, originally designed for discontinuous Galerkin (DG) schemes [42], to the correction procedure via reconstruction (CPR) framework for solving conservation laws. The objective of this simple WENO limiter is to simultaneously maintain uniform high order accuracy of the CPR framework in smooth regions and control ...

The potential of the hybridized discontinuous Galerkin (HDG) method has been recognized for the computation of stationary flows. Extending the method to instationary problems can, e.g., be done by backward difference formulae (BDF) or diagonally implicit Runge-Kutta (DIRK) methods. In this publication, we investigate the use of embedded DIRK methods in an HDG solver, including the use of adapti...

In this paper, we study the superconvergence of the error between the discontinuous Galerkin (DG) finite element solution and the exact solution for linear conservation laws when upwind fluxes are used. We prove that if we apply piecewise k-th degree polynomials, the error between the DG solution and the exact solution is (k+2)-th order superconvergent at the downwind-biased Radau points with s...

We present optimal order preconditioners for certain discontinuous Galerkin (DG) finite element discretizations of elliptic boundary value problems. A specific assembling process is proposed which allows us to use the hierarchy of geometrically nested meshes. We consider two variants of hierarchical splittings and study the angle between the resulting subspaces. Applying the corresponding two-l...

In this paper, we adapt a simple weighted essentially non-oscillatory (WENO) limiter, originally designed for discontinuous Galerkin (DG) schemes on two-dimensional unstructured triangular meshes [35], to the correction procedure via reconstruction (CPR) framework for solving nonlinear hyperbolic conservation laws on two-dimensional unstructured triangular meshes with straight edges or curved e...

We show in this paper how to properly discretize optimized Schwarz methods for the time-harmonic Maxwell equations using a discontinuous Galerkin (DG) method. Due to the multiple traces between elements in the DG formulation, it is not clear a priori how the more sophisticated transmission conditions in optimized Schwarz methods should be discretized, and the most natural approach does not lead...

In this work we propose a hybridizable discontinuous Galerkin (hdG) discretization of the high-frequency Helmholtz equation in the presence of point sources and highly heterogeneous and discontinuous wave speed models. We show that it delivers solutions that are provably second-order accurate and do not suffer from the pollution error, as long as a slightly higher order hdG method is used where...

We construct optimal order multilevel preconditioners for interiorpenalty discontinuous Galerkin (DG) finite element discretizations of 3D elliptic boundary-value problems. A specific assembling process is proposed which allows us to characterize the hierarchical splitting locally. This is also the key for a local analysis of the angle between the resulting subspaces. Applying the corresponding...

A general framework of constructing C discontinuous Galerkin (CDG) methods is developed for solving the Kirchhoff plate bending problem, following some ideas in [10, 12]. The numerical traces are determined based on a discrete stability identity, which lead to a class of stable CDG methods. A stable CDGmethod, called the LCDGmethod, is particularly interesting in our study. It can be viewed as ...