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

广义不可压缩流动问题的高效求解器，特别强调线性纳维-斯托克斯方程和扩展的压力舒尔补技术

• 批准号: 19206589
• 负责人: Professor Dr. Stefan Turek
• 金额: --
• 依托单位: Lehrstuhl III: Angewandte Mathematik und Numerik
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2005
• 资助国家: 德国
• 起止时间: 2004-12-31 至 2009-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/19206589?language=en
• 关键词: Efficient; solvers; generalized; incompressible; flow

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.

近年来，大量的工作用于解决方案的问题，以鞍点形式出现。在工程领域中，诸如流体和Solidhanics之类的领域是UCH鞍点系统的主要来源，因为它们通常是由不可压缩的流量问题披露的，例如Navier-Stokes方程式。作为一般形式。这项调查表明，案件问题或多或少解决了avier-stokes方程，包括对流零件（OSEEN方程），尤其具有额外数​​量的Stokes方程（k€ E Turberence模型，Boussinesq方程，多相现象，粘弹性杂物，流体结构相互作用）在这个共同项目中的数值流量模拟领域的重要任务。不可压缩的液体，尤其是改善解决方案的策略，我们将专注于非恒定的流动问题。 r基于贴片的定义，将其压力补充，将其用于稳定的符合和不合格的stokes Elemets Arious多项式空间。教养学并实现他们中的求解器组件包装fartflow iCh可以直接验证和评估化学工程应用领域中广泛的典型流量配置。