Efficient solvers for generalized incompressible flow problems with special emphasis on pressure Schur complement techniques for linearized Navier-Stokes equations and extensions

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In recent years, a large amount of work has been devoted to the problem of solving large (linear) systems in saddle point form. The reason for this interest is the fact that such problems arise in a wide variety of technical and scientific applications. In particular, the increasing popularity of mixed finite element methods in engineering fields such as fluid and solid mechanics has been a major source of such saddle point systems, as they typically arise from, the discretization of incompressible flow problems, for instance described by the Navier-Stokes equations. Because of the ubiquitous nature of saddle point systems, a wide literature exists on the discretization aspects and the numerical solution of such systems for many particular applications as well as in general form. A recent comprehensive survey [M. Benzi, G.H. Golub, J. Liesen, Numerical solution of saddle point problems, Acta Numerica 2005, pp.1-137] can serve as an introduction to the subject, where one can find enormous pointers to the literature on numerics for saddle point problems. This survey shows that the case of stationary and time-dependent Stokes problems has been more or less solved, while the development of efficient solvers for linearized Navier-Stokes equations including convective parts (Oseen equations) and particularly nonlinear viscosity (nonnewtonian, resp., granular flow), and moreover also for extensions which couple the Navier-Stokes equations with additional quantities (k ¿ e turbulence models, Boussinesq equations, multiphase phenomena, viscoelastic problems, fluid-structure interaction), is still a challenging and important task in the field of numerical flow simulation. In this common project, we will combine the special knowledge from each of both research groups, regarding theoretical as well as algorithmic aspects for the numerical treatment of incompressible fluids, with the aim to develop, to analyse and to implement improved solution strategies. In particular, we will concentrate on flow problems with non-constant, resp., nonlinear viscosity for small up to medium Re numbers as they typically arise in micro devices and milli-reactors. The main solution methodology will be based on pressure Schur complement techniques, which are either constructed via globally defined approximate preconditioners in the pressure space only, or which are based on patch wisely defined operators including the pressure Schur complement of the complete flow equations, but in a local sense. These approaches will be applied to saddle point problems arising from the FEM discretization with stable conforming as well as nonconforming Stokes elements, including various polynomial spaces. We will theoretically analyse the developed solution ethodology and realize the solver components in the FEM package FEATFLOW which directly allows a validation and evaluation for a wide class of prototypical flow configurations in the field of chemical engineering applications.