Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics
Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics
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DOI：
10.1090/s0025-5718-06-01928-4
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发表时间：
2006-09
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影响因子：
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通讯作者：
Barry Lee
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文献类型：
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作者：
Barry Lee
不可压缩流体动力学的高效数值处理在计算数学中是一项艰巨的挑战。要开发一种高效的处理方法，需要了解流体动力学的基本方程，即斯托克斯方程和纳维 - 斯托克斯方程，以及能使问题适定的边界条件。其次，对于变分形式，需要了解稳定的有限元离散化方法或有限元稳定化技术，以便生成精确的数值解。最后，需要掌握线性和非线性的快速迭代求解方案，这样才能构建一种整体高效的方法。在本书出版之前，要了解这样的前沿进展，主要参考资料是关于这些处理组件中每一个的技术期刊文章。埃尔曼、西尔维斯特和沃森所著的《有限元与快速迭代求解器：在不可压缩流体动力学中的应用》现在在一本书中对最先进的技术进行了全面的介绍。作者们在这个研究领域备受赞誉。实际上，这本书呈现了他们自己的许多研究成果，但以一种专业人士和工程、数学以及跨学科计算科学的学生都能读懂的形式呈现。这本书内容自成体系。只需要偏微分方程离散化方法、基础泛函分析和计算线性代数的基本知识。对流体动力学基础偏微分方程、有限元离散化以及迭代线性系统方案的介绍从基础知识开始。然后，本书系统地推进到不可压缩流体动力学更高级的理论和计算技术。这些高级主题包括连续和离散的inf - sup稳定性以及满足这些稳定性的有限元，稳定化有限元技术和用于验证稳定性的分片检验，克雷洛夫迭代方法及其背后的理论，斯托克斯方程和纳维 - 斯托克斯方程有效预条件子的构建和合理性证明，以及后验自适应过程。尽管这些主题通常只有该领域的专家才能理解，但作者给出了一种易于理解且全面的介绍，通过具体的测试套件示例以及适当的分析和计算练习，让读者掌握这些概念。
The efficient numerical treatment of incompressible fluid dynamics is a formidable challenge in computational mathematics. To develop an efficient treatment, one needs to understand the basic equations of fluid dynamics; i.e., the Stokes and Navier–Stokes equations, together with the boundary conditions that give wellposedness. Next, for variational formulations, one needs a knowledge of finite element discretizations that are stable or finite element stabilization techniques, in order to generate accurate numerical solutions. Finally, one needs a grasp of fast iterative solution schemes, both linear and nonlinear, so that an overall efficient method can be constructed. To gain acquaintance with such a cutting-edge development, prior to the publication of this book, the main reference sources were technical journal articles on each of these treatment components. Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics, by Elman, Silvester, and Wathen, now gives a thorough presentation of the state-ofthe-art techniques in one volume. The authors are acclaimed in this research field. Indeed, this book presents many of their own research results, but in a readable form accessible to a broad audience of both professionals and students of engineering, mathematics, and interdisciplinary computational science. The book is self-contained. Only basic knowledge of discretization methods for partial differential equations, fundamental functional analysis, and computational linear algebra are needed. Presentation of the basic partial differential equations underlying fluid dynamics, of finite element discretization, and of iterative linear system schemes start from the basics. The book then systematically progresses to more advanced theoretical and computational techniques for incompressible fluid dynamics. These advanced topics include continuous and discrete inf-sup stability and finite elements that satisfy them, stabilized finite element techniques and the patch-test to verify stability, Krylov iterative methods and the theory behind them, construction and justification of efficient preconditioners for the Stokes and Navier–Stokes equations, and a posteriori adaptive procedures. Even though these topics are often accessible only to experts in the field, the authors give a comprehendible yet thorough presentation, with concrete test-suite examples and appropriate analytical and computational exercises to give the reader a grasp of these concepts.