Stabilization strategies for discontinuous Galerkin methods

The subject of this thesis is the analysis of discontinuous Galerkin methods for linear partial differential equations of first or second order. Discontinuous Galerkin methods are known to satisfy a local mass conservation property. Taking a closer look one can observe that this property often depends on the method itself, i.e. on some method-dependent parameters. The main objective of this work is to design discontinuous Galerkin schemes satisfying a method-independent local mass conservation property still ensuring full stability and optimal convergence of the approximations. Depending on the problem and the choice of the approximation space, the strategy to reach this goal might be different. We give a precise characterization of what type of stabilization term is needed in order to obtain a stable and optimally convergent numerical scheme. Both high order and low order approximation spaces are treated. We present strategies for the scalar hyperbolic, first and second order elliptic, parabolic problems and the Stokes equations. In each case we establish an analysis proving stability and optimal convergence of the approximations. Some numerical examples illustrate the theoretical results.