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，大量工作致力于解决鞍点形式的大型（线性）系统问题。引起这种兴趣的原因是这样的问题在各种技术和科学应用中都会出现。特别是，混合有限元方法在流体和固体力学等工程领域中的日益普及，已成为此类鞍点系统的主要来源，因为它们通常源自不可压缩流动问题的离散化，例如纳维-斯托克斯方程所描述的问题。由于鞍点系统的普遍性，存在大量关于此类系统的离散化方面和数值解的文献，适用于许多特定应用以及一般形式。最近的一项综合调查[M.本齐，G.H. Golub, J. Liesen，鞍点问题的数值解，Acta Numerica 2005，第 1-137 页] 可以作为该主题的介绍，其中可以找到大量有关鞍点问题的数值文献。这项调查表明，平稳和瞬态斯托克斯问题的情况已或多或少得到解决，同时开发了线性纳维-斯托克斯方程的有效求解器，包括对流部分（Oseen 方程）和特别是非线性粘度（非牛顿、颗粒流），此外还用于将纳维-斯托克斯方程与附加量耦合的扩展（k ¿ e 湍流模型、Boussinesq） 方程、多相现象、粘弹性问题、流固耦合），仍然是数值流动模拟领域中具有挑战性和重要的任务。在这个共同项目中，我们将结合两个研究小组在不可压缩流体数值处理的理论和算法方面的专业知识，旨在开发、分析和实施改进的解决策略。特别是，我们将重点关注小到中等 Re 数的非恒定非线性粘度流动问题，因为它们通常出现在微型设备和毫反应器中。主要的求解方法将基于压力 Schur 补技术，该技术要么仅通过压力空间中全局定义的近似预处理器构建，要么基于补丁明智定义的算子，包括完整流动方程的压力 Schur 补，但在局部意义上。这些方法将应用于由具有稳定一致和非一致斯托克斯元素（包括各种多项式空间）的有限元离散化引起的鞍点问题。我们将从理论上分析所开发的解决方案方法，并在 FEM 包 FEATFLOW 中实现求解器组件，该组件可以直接对化学工程应用领域中的各种原型流动配置进行验证和评估。