A posteriori estimates for the Bubble Stabilized Discontinuous Galerkin Method

with f ∈ L2(Ω), a reaction coefficient τ > 0 and a diffusion coefficient that is piecewise constant on each element and satisfies ε(x) > ε0 > 0. We assume that there exists a constant ρ > 0 such that ε|κ1 6 ρε|κ2 for two elements satisfying ∂κ1 ∩ ∂κ2 6= / 0, i.e. in other words that ε is of bounded variation from one element to the other. The Bubble Stabilized Discontinuous Galerkin (BSDG) was first developed for Poissons’s problem by Brezzi and Marini (2006) for the non-symmetric formulation and by Burman and Stamm (2008c) for both the symmetric and non-symmetric variants. A more refined analysis was then presented by Burman and Stamm (2008b). In Burman and Stamm (2008a) the method was extended to the diffusion-reaction problem as described by equation (1.1) and to time dependent problems. Further, superconvergence of some residual quantities, that play an important role in the upcoming a posteriori analysis, are pointed out. In addition, the BSDG-method has a close relation to the classical mixed lowest order RaviartThomas method. A posteriori estimations for discontinuous Galerkin methods is a recent and fast developing research area. First results were published by Karakashian and Pascal (2003); Rivière and Wheeler (2003) and Becker et al. (2003). A posteriori estimates are mostly used for problems with lower regularity of the exact solution, i.e. u ∈ H1(Ω) in order to solve problems where a local refinement strategy is really needed. Therefore, the theory of a posteriori estimates was further developed in (Ainsworth, 2007; Ern et al., 2008; Houston et al., 2007, 2008; Stephansen, 2007) to provide estimates that are firstly build on the assumption of u ∈ H1(Ω), instead of u ∈ H2(Ω) as in some of the earliest works. Secondly, attention is given to have a better and if possible an explicit control of the constants. A posteriori estimates with strongly variable diffusions coefficients are discussed by Ern and Stephansen (2008) using the technique of weighted averages. Based on a posteriori estimates, adaptive refinement strategies were designed by Hoppe et al. (2008); Karakashian and Pascal (2007); Bonito and Nochetto (2008) and global convergence towards the exact solution can be proven.