Finite Element Methods And Their Applications
Finite Element Methods And Their Applications
复制标题
DOI：
10.1007/3-540-28078-2
复制
发表时间：
2005
期刊：
影响因子：
--
通讯作者：
Zhangxin Chen
中科院分区：
文献类型：
--
作者：
Zhangxin Chen
二十年前，当我们撰写流体动力学谱方法（1988）时，这个主题仍然相当新颖。受到我们收到的许多好评以及对该书（将被称为 CHQZ1）的持续兴趣的激励，但又希望提出一个更现代的视角，我们开始了这个项目，最终产生了我们最近的书（Canuto 等人（2006），被称为 CHQZ2）和现在的新书（被称为 CHQZ3）。这两本新书的目标是使我们对经典谱方法的深入讨论现代化，考虑到流体动力学领域的理论进步和更广泛的应用经验，同时从经典谱方法的角度总结多域谱方法的现状。虽然这两本新书大量借鉴了我们早期教材的部分内容，但 CHQZ1、CHQZ2 的大部分内容以及 CHQZ3 的大部分内容都是新的。添加的内容使得我们有必要将这部新作品作为两本书出版。在各本书之间划分材料的理由是，我们在第一本新书中对单一领域中经典谱方法的基本方面进行了全面的讨论。第二本新书重点关注流体动力学和多域谱方法的应用。 CHQZ2 的前言中详细介绍了谱方法从最初（现在的经典）版本到当代多域版本的历史演变。简而言之，用于解决光滑问题的经典谱方法的理论和算法在 20 世纪 80 年代中期就已经相当成熟，并且在过去二十年中在将谱方法扩展到任意几何形状方面取得了非凡的进展，使得某些人认为任意高阶方法能够应用于任意几何问题的数学涅槃。在这方面，谱法在过去20年的发展轨迹已经接近hp有限元法。从单域谱方法到多域谱方法的迁移过程需要注入新的数学工具并激发原始的研究方向。数学对方法的正确设计和解释产生了深远的影响，在某些情况下，它甚至启发了不连续谱方法的发展，即使对于连续解的问题也是如此。另一方面，由于一般来说，几何
Two decades ago when we wrote Spectral Methods in Fluid Dynamics (1988), the subject was still fairly novel. Motivated by the many favorable comments we have received and the continuing interest in that book (which will be referred to as CHQZ1), and yet desiring to present a more modern perspective, we embarked on the project which resulted in our recent book (Canuto et al.(2006), referred to as CHQZ2) and the present new book (referred to as CHQZ3). Our objectives with these two new books are to modernize our thorough discussion of classical spectral methods, accounting for advances in the theory and more extensive application experience in the fluid dynamics arena, while summarizing the current state of multidomain spectral methods from the perspective of classical spectral methods. While the two new books draw extensively from portions of our earlier text, CHQZ1, much of CHQZ2, and most of CHQZ3 is new. The added content has necessitated our publishing this new work as two separate books. The rationale for the division of the material between the books is that we furnished in the first new book a comprehensive discussion of the fundamental aspects of classical spectral methods in single domains. This second new book focuses on applications to fluid dynamics and on multidomain spectral methods. The historical evolution of spectral methods from their initial (now classical) versions to the contemporary multidomain versions has been covered in some detail in the Preface of CHQZ2. In short, both the theory and the algorithms of classical spectral methods for smooth problems were reasonably mature already in the mid-1980s, and singular progress has been made over the past two decades in extending spectral methods to arbitrary geometries, enabling what some would consider the mathematical nirvana of a method of arbitrarily high-order capable of application to problems on an arbitrary geometry. In this respect, the trajectory of spectral methods over the past 20 years has been approaching that of hp finite-element methods. This process of migration from single-domain to multidomain spectral methods has required the injection of novel mathematical tools and stimulated original investigation directions. Mathematics has had a profound impact on the correct design and interpretation of the methods, and in some cases it has inspired the development of discontinuous spectral methods even for problems with continuous solutions. On the other hand, since in general a geometrically