Fully Computable Error Bounds for Discontinuous Galerkin Finite Element Approximations on Meshes with an Arbitrary Number of Levels of Hanging Nodes

Abstract

We obtain fully computable a posteriori error bounds on the broken energy seminorm and discontinuous Galerkin norm (DG-norm) of the error in first order symmetric interior penalty Galerkin (SIPG), nonsymmetric interior penalty Galerkin (NIPG), and incomplete interior penalty Galerkin (IIPG) finite element approximations of a linear second order elliptic problem on meshes containing an arbitrary number of levels of hanging nodes and comprised of triangular elements. The estimators are completely free of unknown constants and provide guaranteed numerical bounds on the broken energy seminorm and DG-norm of the error. These estimators are also shown to provide a lower bound for the broken energy seminorm and DG-norm of the error up to a constant and higher order data oscillation terms. We also obtain an explicit computable bound for the value of the interior penalty parameter needed to ensure the existence of the discontinuous Galerkin finite element approximation for all versions of the method.