Least-Squares Finite Element Methods (eBook)
XXII, 660 Seiten
Springer New York (Verlag)
978-0-387-68922-7 (ISBN)
Since their emergence, finite element methods have taken a place as one of the most versatile and powerful methodologies for the approximate numerical solution of Partial Differential Equations. These methods are used in incompressible fluid flow, heat, transfer, and other problems. This book provides researchers and practitioners with a concise guide to the theory and practice of least-square finite element methods, their strengths and weaknesses, established successes, and open problems.
Since their emergence in the early 1950s, ?nite element methods have become one of the most versatile and powerful methodologies for the approximate numerical solution of partial differential equations. At the time of their inception, ?nite e- ment methods were viewed primarily as a tool for solving problems in structural analysis. However, it did not take long to discover that ?nite element methods could be applied with equal success to problems in other engineering and scienti?c ?elds. Today, ?nite element methods are also in common use, and indeed are often the method of choice, for incompressible ?uid ?ow, heat transfer, electromagnetics, and advection-diffusion-reaction problems, just to name a few. Given the early conn- tion between ?nite element methods and problems engendered by energy minimi- tion principles, it is not surprising that the ?rst mathematical analyses of ?nite e- ment methods were given in the environment of the classical Rayleigh-Ritz setting. Yet again, using the fertile soil provided by functional analysis in Hilbert spaces, it did not take long for the rigorous analysis of ?nite element methods to be extended to many other settings. Today, ?nite element methods are unsurpassed with respect to their level of theoretical maturity.
Preface 7
Contents 15
Part I Survey of Variational Principles and Associated Finite Element Methods 23
Classical Variational Methods 24
Alternative Variational Formulations 55
Part II Abstract Theory of Least-Squares Finite Element Methods 86
Mathematical Foundations of Least-Squares Finite Element Methods 87
The Agmon–Douglis–Nirenberg Setting for Least-Squares Finite Element Methods 120
Part III Least-Squares Finite Element Methods for Elliptic Problems 148
Scalar Elliptic Equations 149
Vector Elliptic Equations 213
The Stokes Equations 253
Part IV Least-Squares Finite Element Methods for Other Settings 325
The Navier–Stokes Equations 326
Parabolic Partial Differential Equations 381
Hyperbolic Partial Differential Equations 417
Control and Optimization Problems 443
Variations on Least-Squares Finite Element Methods 489
Part V Supplementary Material 545
Analysis Tools 546
Compatible Finite Element Spaces 565
Linear Operator Equations in Hilbert Spaces 597
The Agmon–Douglis–Nirenberg Theory and Verifying its Assumptions 604
References 636
Acronyms 652
Glossary 653
Index 656
Erscheint lt. Verlag | 28.4.2009 |
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Reihe/Serie | Applied Mathematical Sciences | Applied Mathematical Sciences |
Zusatzinfo | XXII, 660 p. |
Verlagsort | New York |
Sprache | englisch |
Themenwelt | Mathematik / Informatik ► Mathematik ► Allgemeines / Lexika |
Mathematik / Informatik ► Mathematik ► Analysis | |
Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik | |
Technik ► Bauwesen | |
Schlagworte | Analysis • Bochev • Elements • Finite • Finite Element Method • hyperbolic partial differential equation • Least-Squares • linear optimization • Operator • Optimization |
ISBN-10 | 0-387-68922-2 / 0387689222 |
ISBN-13 | 978-0-387-68922-7 / 9780387689227 |
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