Title: A Massively Parallel Finite Element Framework with Application to Incompressible Flows
Abstract: The incompressible Navier-Stokes equations describe the flow of Newtonian fluids or gases. They are used in many numerical simulations of flow problems. These simulations have become an important task in research and industrial applications.In the recent years, one can observe a trend regarding flow simulations: the demand for higher accuracy in the numerical resolution with the need to get the results faster. The underlying geometry is getting more complex and the accuracy requirements ask for discretizing with smaller and smaller cells. All this results in a dramatic increase in the problem size -- and therefore computational complexity and memory requirements -- that must be solved. Additionally, to achieve higher accuracy more elaborate physical models will be required in order to capture more phenomena from the real problem.To cope with this trend, one has basically two options: First, increase the computational power, i.e., using faster and bigger computers. The requirements can typically only be met by parallel machines. In return, the finite element software must be designed to explicitly to run efficiently on a parallel machine. Second, one can try to improve the efficiency of the algorithms and solvers.This thesis combines these two approaches by deriving, implementing, and verifying a massively parallel, generic finite element framework with adaptive mesh refinement and by developing a fast and robust solver for the linear saddle point problems arising from the discretization of incompressible flow problems taking advantage of a stabilization term known as Grad-Div stabilization.