Convergence of adaptive discontinuous Galerkin approximations of second-order elliptic problems

A 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. An adaptive algorithm using this estimator together with specific marking and refinement 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. Results of numerical experiments are presented.