Discontinuous Galerkin Method For Elliptic Problems Research Articles

A discontinuous Galerkin discretization of elliptic problems with improved convergence properties using summation by parts operators

Nishikawa (2007) proposed to reformulate the classical Poisson equation as a steady state problem for a linear hyperbolic system. This results in optimal error estimates for both the solution of the elliptic equation and its gradient. However, it prevents the application of well-known solvers for elliptic problems. We show connections to a discontinuous Galerkin (DG) method analyzed by Cockburn, Guzmán, and Wang (2009) that is very difficult to implement in general. Next, we demonstrate how this method can be implemented efficiently using summation by parts (SBP) operators, in particular in the context of SBP DG methods such as the DG spectral element method (DGSEM). The resulting scheme combines nice properties of both the hyperbolic and the elliptic point of view, in particular a high order of convergence of the gradients, which is one order higher than what one would usually expect from DG methods for elliptic problems.

Discontinuous Galerkin Finite Element Methods for Interface Problems: A Priori and A Posteriori Error Estimations

Discontinuous Galerkin (DG) finite element methods were studied by many researchers for second-order elliptic partial differential equations, and a priori error estimates were established when the solution of the underlying problem is piecewise $H^{3/2+\epsilon}$ smooth with $\epsilon>0$. However, elliptic interface problems with intersecting interfaces do not possess such a smoothness. In this paper, we establish a quasi-optimal a priori error estimate for interface problems whose solutions are only $H^{1+\alpha}$ smooth with $\alpha\in(0,1)$ and, hence, fill a theoretical gap of the DG method for elliptic problems with low regularity. The second part of the paper deals with the design and analysis of robust residual- and recovery-based a posteriori error estimators. Theoretically, we show that the residual and recovery estimators studied in this paper are robust with respect to the DG norm, i.e., their reliability and efficiency bounds do not depend on the jump, provided that the distribution of coefficients is locally quasi-monotone.

Adjoint Consistency Analysis of Discontinuous Galerkin Discretizations

This paper is concerned with the adjoint consistency of discontinuous Galerkin (DG) discretizations. Adjoint consistency—in addition to consistency—is the key requirement for DG discretizations to be of optimal order in $L^2$ as well as measured in terms of target functionals. We provide a general framework for analyzing the adjoint consistency of DG discretizations which is also useful for the derivation of adjoint consistent methods. This analysis will be performed for the DG discretizations of the linear advection equation, the interior penalty DG method for elliptic problems, and the DG discretization of the compressible Euler equations. This framework is then used to derive an adjoint consistent DG discretization of the compressible Navier-Stokes equations. Numerical experiments demonstrate the link of adjoint consistency to the accuracy of numerical flow solutions and the smoothness of discrete adjoint solutions.