Optimal Convergence Rates for Adaptive Lowest-Order Discontinuous Petrov--Galerkin Schemes

Convergence Analysis of the Lowest Order Weakly Penalized Adaptive Discontinuous Galerkin Methods

In this article, we prove convergence of the weakly penalized adaptive discontinuous Galerkin methods. Unlike other works, we derive the contraction property for various discontinuous Galerkin methods only assuming the stabilizing parameters are large enough to stabilize the method. A central idea in the analysis is to construct an auxiliary solution from the discontinuous Galerkin solution by ...

Quasi-Optimal Convergence Rate of an Adaptive Discontinuous Galerkin Method

We analyze an adaptive discontinuous finite element method (ADFEM) for symmetric second order linear elliptic operators. The method is formulated on nonconforming meshes made of simplices or quadrilaterals, with any polynomial degree and in any dimension ≥ 2. We prove that the ADFEM is a contraction for the sum of the energy error and the scaled error estimator, between two consecutive adaptive...

A-stable discontinuous Galerkin-Petrov time discretization of higher order

Arbitrary High Order Discontinuous Galerkin Schemes

In this paper we apply the ADER one step time discretization to the Discontinuous Galerkin framework for hyperbolic conservation laws. In the case of linear hyperbolic systems we obtain a quadrature-free explicit single-step scheme of arbitrary order of accuracy in space and time on Cartesian and triangular meshes. The ADERDG scheme does not need more memory than a first order explicit Euler ti...

Adaptive Petrov-Galerkin Methods for First Order Transport Equations

We propose a general framework for well posed variational formulations of linear unsymmetric operators, taking first order transport and evolution equations in bounded domains as primary orientation. We outline a general variational framework for stable discretizations of boundary value problems for these operators. To adaptively resolve anisotropic solution features such as propagating singula...