On the Design of Global-in-Time Newton-Multigrid-Pressure Schur Complement Solvers for Incompressible Flow Problems

IF 16.4 1区 化学 Q1 CHEMISTRY, MULTIDISCIPLINARY
Christoph Lohmann, Stefan Turek
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Abstract

In this work, a new global-in-time solution strategy for incompressible flow problems is presented, which highly exploits the pressure Schur complement (PSC) approach for the construction of a space–time multigrid algorithm. For linear problems like the incompressible Stokes equations discretized in space using an inf-sup-stable finite element pair, the fundamental idea is to block the linear systems of equations associated with individual time steps into a single all-at-once saddle point problem for all velocity and pressure unknowns. Then the pressure Schur complement can be used to eliminate the velocity fields and set up an implicitly defined linear system for all pressure variables only. This algebraic manipulation allows the construction of parallel-in-time preconditioners for the corresponding all-at-once Picard iteration by extending frequently used sequential PSC preconditioners in a straightforward manner. For the construction of efficient solution strategies, the so defined preconditioners are employed in a GMRES method and then embedded as a smoother into a space–time multigrid algorithm, where the computational complexity of the coarse grid problem highly depends on the coarsening strategy in space and/or time. While commonly used finite element intergrid transfer operators are used in space, tailor-made prolongation and restriction matrices in time are required due to a special treatment of the pressure variable in the underlying time discretization. The so defined all-at-once multigrid solver is extended to the solution of the nonlinear Navier–Stokes equations by using Newton’s method for linearization of the global-in-time problem. In summary, the presented multigrid solution strategy only requires the efficient solution of time-dependent linear convection–diffusion–reaction equations and several independent Poisson-like problems. In order to demonstrate the potential of the proposed solution strategy for viscous fluid simulations with large time intervals, the convergence behavior is examined for various linear and nonlinear test cases.

Abstract Image

不可压缩流动问题全局实时牛顿-多网格-压力Schur补解的设计
本文提出了一种新的不可压缩流动问题的全局实时求解策略,该策略高度利用了压力舒尔补(PSC)方法来构建时空多网格算法。对于像不可压缩的Stokes方程这样的线性问题,使用非超稳定有限元对在空间中离散化,其基本思想是将与单个时间步长相关的线性方程组阻塞为所有速度和压力未知量的单一鞍点问题。然后利用压力舒尔补来消除速度场,建立一个只包含所有压力变量的隐式定义线性系统。这种代数操作允许通过以直接的方式扩展常用的顺序PSC前置条件,为相应的一次性皮卡德迭代构建并行时间前置条件。为了构建有效的解策略,将上述预处理条件应用于GMRES方法中,然后将其作为平滑器嵌入到时空多网格算法中,其中粗网格问题的计算复杂度高度依赖于空间和/或时间上的粗化策略。在空间上使用常用的有限元网格间传递算子时,由于在底层时间离散中对压力变量进行了特殊处理,需要在时间上定制扩展和约束矩阵。利用牛顿线性化全局实时问题的方法,将上述一次性多网格求解方法推广到非线性Navier-Stokes方程的求解。综上所述,所提出的多网格求解策略只需要有效地求解随时间变化的线性对流-扩散-反应方程和若干独立的类泊松问题。为了证明所提出的求解策略在大时间间隔粘性流体模拟中的潜力,研究了各种线性和非线性测试用例的收敛性。
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来源期刊
Accounts of Chemical Research
Accounts of Chemical Research 化学-化学综合
CiteScore
31.40
自引率
1.10%
发文量
312
审稿时长
2 months
期刊介绍: Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.
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