hp-DGFEM for nonlinear convection-diffusion problems

We deal with a numerical solution of nonlinear convection-diffusion problems with the aid of the discontinuous Galerkin finite element method (DGFEM). We propose a new hp-adaptation technique, which is based on a combination of a residuum-nonconformity estimator and a regularity indicator. The residuum-nonconformity estimator consists of two building blocks and it marks mesh elements for a refinement. The regularity indicator decides if the marked elements will be refined by h- or p-technique. The residuum-nonconformity estimator as well as the regularity indicator are easily computable quantities. Moreover, the same technique estimates an algebraic error arising from an iterative solution of the corresponding nonlinear algebraic system. The performance of the proposed hp-DGFEM is demonstrated by five numerical examples.