Analysis of hybrid discontinuous Galerkin methods for incompressible flow problems

In this work, we consider the derivation and analysis of finite element methods for the approximate solution of the Navier-Stokes equations, which describe the motion of liquid and gaseous substances. We investigate the stability and approximation properties of primal hybrid discontinuous Galerkin methods, which, due to their local conservation properties and the uncomplicated stabilization of convective transport terms, render themselves as promising candidates for the solution of flow problems. Moreover, the hybrid discontinuous approach facilitates the elimination of local degrees of freedom, the implementation of locally adaptive solvers, and allows for a high level of parallelism. In general, smooth parts of the solution can be efficiently approximated by higher order polynomials, whereas in regions of lower smoothness, e.g., in the vicinity of reentrant corners, one can expect better results by using locally refined meshes. Besides the approximation properties of the underlying ansatz spaces, the upper bound for the discretization error considerably depends on certain constants appearing in the stability bounds. Hence, we explicitly keep track of the dependence of these constants with respect to the (local) mesh-size and the (local) polynomial degree of approximation. The analysis includes meshes with hanging nodes, consisting of different types of elements. Subsequently to the analysis of the method, we derive and analyze efficient and reliable error estimators. These error estimators allow to explicitly compute bounds for the solution and can be used as local indicators to assess the regions in the computational domain, where we can expect a significant improvement of the approximation properties by a local refinement. Such error estimators are an essential ingredient for the implementation and analysis of adaptive solution algorithms. Again, we make the dependence of the lower and upper error bounds explicit with respect to the polynomial degree. Finally, we demonstrate how hybridization techniques can be employed to couple interface problems between conforming finite element discretizations on subdomains. The resulting hybrid mortar methods can be embedded into the framework of domain decomposition algorithms. Although the focus of this work is on numerical analysis, we illustrate our theoretical findings with numerical results and shortly discuss implementation issues