Adaptive Finite Element Methods for Computing Nonstationary Incompressible Flows

Abstract

Subject of this work is the development of numerical methods for efficiently solving nonstationary incompressible flow problems. In contrast to stationary flow problems, here errors due to discretization in time and space occur. Furthermore, especially three-dimensional simulations lead to huge computational costs. Thus, adaptive discretization methods have to be used in order to reduce the computational costs while still maintaining a certain accuracy. The main focus of this thesis is the development of an a posteriori error estimator which is computable and able to assess both discretization errors separately. Thereby, the error is measured in an arbitrary quantity of interest (such as the drag-coefficient, for example) because measuring errors in global norms is often of minor importance in practical applications. The basis for this is a finite element discretization in time and space. The techniques presented here also provide local error indicators which are used to adaptively refine the temporal and spatial discretization. A key ingredient in setting up an efficient discretization method is balancing the error contributions due to temporal and spatial discretization. To this end, a quantitative assessment of the individual discretization errors is required. The described methods are validated by several numerical tests. These also include established Navier-Stokes benchmarks as well as a two-phase flow problem with complex three-dimensional geometry.