ENERGY NORM A POSTERIORI ERROR ESTIMATION OF hp-ADAPTIVE DISCONTINUOUS GALERKIN METHODS FOR ELLIPTIC PROBLEMS

Abstract

In this paper, we develop the a posteriori error estimation of hp-version interior penalty discontinuous Galerkin discretizations of elliptic boundary-value problems. Computable upper and lower bounds on the error measured in terms of a natural (mesh-dependent) energy norm are derived. The bounds are explicit in the local mesh sizes and approximation orders. A series of numerical experiments illustrate the performance of the proposed estimators within an automatic hp-adaptive refinement procedure.

Second-order elliptic PDEs with discontinuous boundary data

“…This effect has been observed earlier in the a posteriori error analysis of hp-methods; see, e.g., Melenk & Wohlmuth (2001) and Houston et al (2007Houston et al ( , 2008. We remark, however, that the p-suboptimality is less pronounced in energy norm lower bounds.…”

“…Arnold et al (2001), and the references cited therein. DG methods are ideally suited for realizing hp-adaptivity for secondorder boundary value problems, an advantage that has been noted early on in the recent development of these methods; see, for example, Baumann & Oden (1999), Rivière et al (1999), Cockburn et al (2000), Perugia & Schötzau (2002), Wihler et al (2003), Houston et al (2002Houston et al ( , 2007Houston et al ( , 2008, Stamm & Wihler (2010) and the references therein. Indeed, working with discontinuous finite element spaces easily facilitates the use of variable polynomial degrees and local mesh refinement techniques on possibly irregularly refined meshes-the two key ingredients for hp-adaptive algorithms.…”

An a posteriori error indicator for discontinuous Galerkin approximations of fourth-order elliptic problems

“…also [7,21,20] for similar constructions). Using this recovery operator, in conjunction with the inconsistent formulation for the IPDG presented in [17] (which ensures that the weak formulation of the problem is defined under minimal regularity assumptions on the analytical solution), we derive efficient and reliable a posteriori estimates of residual type for the IPDG method in the corresponding energy norm.…”

“…Using this recovery operator, in conjunction with the inconsistent formulation for the IPDG presented in [17] (which ensures that the weak formulation of the problem is defined under minimal regularity assumptions on the analytical solution), we derive efficient and reliable a posteriori estimates of residual type for the IPDG method in the corresponding energy norm. Some ideas from a posteriori analyses for the Poisson problem presented in [4,21,20,1,9] are also implicitly utilized here in the context of fourth order problems.…”

Discontinuous Galerkin finite element approximation of quasilinear elliptic boundary value problems II: strongly monotone quasi-Newtonian flows

“…This general approach was pursued in the series of articles by Karakashian & Pascal (2003) and Houston et al (2004aHouston et al ( , 2005bHouston et al ( , 2007Houston et al ( , 2008. The proof of the local lower error bounds (efficiency) is based on the techniques presented in Melenk & Wohlmuth (2001), subject to the treatment of the nonlinearity.…”