discontinuous Galerkin method; $hp$-adaptation; residuum estimator; regularity indicator; mesh refinement; stationary nonlinear convection-diffusion problems; error estimation; numerical experiments
We deal with a numerical solution of nonlinear convection-diffusion equations with the aid of the discontinuous Galerkin method (DGM). We propose a new $hp$-adaptation technique, which is based on a combination of a residuum estimator and a regularity indicator. The residuum estimator as well as the regularity indicator are easily evaluated quantities without the necessity to solve any local problem and/or any reconstruction of the approximate solution. The performance of the proposed $hp$-DGM is demonstrated.