Abstract: In this work,  Fully computable convergence analysis 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 is obtained on meshes containing an arbitrary number of levels
of hanging nodes and comprised of triangular elements. We use an estimator which is
completely free of unknown constants and provides a guaranteed numerical bound on
the broken energy norm of the error. This residual-type a posteriori error estimator is
introduced and analyzed for a discontinuous Galerkin formulation of a model second order
elliptic problem with Dirichlet Neumann-type boundary conditions in Rankin's thesis
 (University of Strathclyde).
An adaptive algorithm using this estimator together with specifi c marking and refi nement
strategies is constructed and shown to achieve any specified error level in the energy
norm in a finite number of cycles. The convergence rate is in effect linear with a
guaranteed error reduction at every cycle.