Robust a posteriori error estimation for discontinuous Galerkin methods for convection diffusion problems

The present thesis is concerned with the development and practical implementation of robust a-posteriori error estimators for discontinuous Galerkin (DG) methods for convection-diffusion problems. It is well-known that solutions to convection-diffusion problems may have boundary and internal layers of small width where their gradients change rapidly. A powerful approach to numerically resolve these layers is based on using hp-adaptive finite element methods, which control and minimize the discretization errors by locally adapting the mesh sizes (h-refinement) and the approximation orders (p-refinement) to the features of the problems. In this work, we choose DG methods to realize adaptive algorithms. These methods yield stable and robust discretization schemes for convection-dominated problems, and are naturally suited to handle local variations in the mesh sizes and approximation degrees as required for hp-adaptivity. At the heart of adaptive finite element methods are a-posteriori error estimators. They provide information on the errors on each element and indicate where local refinement/derefinement should be applied. An efficient error estimator should always yield an upper and lower bound of the discretization error in a suitable norm. For convection-diffusion problems, it is desirable that the estimator is also robust, meaning that the upper and lower bounds differ by a factor that is independent of the mesh Peclet number of the problem. We develop a new approach to obtain robust a-posteriori error estimates for convection-diffusion problems for h-version and hp-version DG methods. The main technical tools in our analysis are new hp-version approximation results of an averaging operator, which are derived for irregular hexahedral meshes in three dimensions, as well as for irregular anisotropic rectangular meshes in two dimensions. We present a series of numerical examples based on C++ implementations of our methods. The numerical results indicate that the error estimator is effective in locating and resolving interior and boundary layers. For the hp-adaptive algorithms, once the local mesh size is of the same order as the width of boundary or interior layers, both the energy error and the error estimator are observed to converge exponentially fast in the number of degrees of freedom.Science, Faculty ofMathematics, Department ofGraduat