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

• Applicant: Professor Dr. Stefan Turek
• Subject Area: Mathematics
• Term: from 2005 to 2010
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 19206589

• Final Report Year: 2012

Final Report Abstract

In recent years, a large amount of work has been devoted to the problem of solving large (nonlinear) 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 and extensions. Because of the ubiquitous nature of saddle point systems, a wide literature existed already during beginning of the project on the discretization aspects and the numerical solution of such systems for many particular applications as well as in general form. However, the state-of-theart for the incompressible Navier-Stokes equations could be summarized in such a way that the case of stationary and time-dependent Stokes problems has been more or less solved, while the development of robust and efficient solvers for linearized Navier-Stokes equations including convective parts (Oseen equations) and particularly nonlinear viscosity (non-newtonian fluids), and moreover also for extensions which couple the Navier-Stokes equations with additional quantities (Boussinesq equations, multiphase phenomena, viscoelastic problems, fluid-structure interaction, optimal control), has been and is still a challenging and important task in the field of numerical flow simulation. In this common project (with Maxim Olshanskii/Moscow), we have combined 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 have concentrated on flow problems with non-constant, resp., nonlinear viscosity for small up to medium Re numbers as they typically arise in many applications. The main solution methodology is 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 patchwisely defined operators including the pressure Schur complement of the complete flow equations, but in a local sense. These approaches have been applied to saddle point problems arising from the FEM discretization of the Navier-Stokes equations with stable conforming as well as nonconforming Stokes elements, including various polynomial spaces. We have theoretically and numerically analysed the developed solution methodology and have realized the corresponding solver components in the FEM package FEATFLOW so that directly a validation and evaluation for a wide class of prototypical flow configurations could be performed. Currently, the developed methodology has been extended to optimal control and fluid-structure-interaction problems, too, and is the basis for several simulation projects granted by the DFG.

Publications

• A discrete projection method for incompressible viscous flow with Coriolis force. Computer Methods in Applied Mechanics and Engineering 2008: 197 (51-52): 4512–4520
  A. Sokolov, M. Olshanskii, and S. Turek
  
• Analysis and Numerical Realisation of Discrete Projection Methods for Rotating Incompressible Flows. PhD thesis 2008, TU Dortmund
  A. Sokolov
  
• Numerical study of a new discrecte projection method for rotating incompressible flows. Electronic Transactions on Numerical Analysis 2008; 32: 49–62
  A. Sokolov, M. Olshanskii, and S. Turek
  
• Two finite-difference schemes for calculation of Bingham fluid flows in a cavity. Russ. J. Numer. Anal. Math. Modelling 2008; 23: 615–634
  E. A. Muravleva and M. A. Olshanskii
  
• A monolithic FEM approach for temperature and shear dependent viscosity in viscoelastic flow. 7th EUROMECH Solid Mechanics Conference, ACM Press, 2009
  H. Damanik, A. Ouazzi, J. Hron, and S. Turek
  
• An iterative method for the Stokes type problem with variable viscosity. SIAM J.Sci.Comp. 2009; 32: 3959–3978
  P. P. Grinevich and M. A. Olshanskii
  
• Analysis of semi-staggered finite-difference method with application to Bingham flows. Computer Methods in Applied Mechanics and Engineering 2009: 198 975–985
  M. A. Olshanskii
  
• Error analysis of a projection method for the Navier– Stokes equations with coriolis force. Journal of Mathematical Fluid Mechanics 2009; 12 (4): 485–502
  M. Olshanskii, A. Sokolov, and S. Turek
  
• A monolithic FEM approach for the log–conformation reformulation (LCR) of viscoelastic flow problems. Journal of Non-Newtonian Fluid Mechanics 2010; 165 (19-20): 1105–1113
  H. Damanik, J. Hron, A. Ouazzi, and S. Turek
  
• Numerical solver for the variable viscosity Stokes type problem and applications. PhD thesis 2010, Moscow State Lomonosov University
  P. P. Grinevich