Overview
- Structured to allow for concise development of ideas in a classroom setting
- Includes chapter-level exercises with solutions available online
- Provides proofs and examples throughout each chapter
Part of the book series: Texts in Applied Mathematics (TAM, volume 73)
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Table of contents (32 chapters)
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Front Matter
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Weak Formulations and Well-Posedness
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Front Matter
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Galerkin Approximation
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Front Matter
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Elliptic PDEs: Conforming Approximation
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Front Matter
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Elliptic PDEs: Nonconforming Approximation
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Front Matter
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Keywords
About this book
This book is the second volume of a three-part textbook suitable for graduate coursework, professional engineering and academic research. It is also appropriate for graduate flipped classes. Each volume is divided into short chapters. Each chapter can be covered in one teaching unit and includes exercises as well as solutions available from a dedicated website. The salient ideas can be addressed during lecture, with the rest of the content assigned as reading material. To engage the reader, the text combines examples, basic ideas, rigorous proofs, and pointers to the literature to enhance scientific literacy.
Volume II is divided into 32 chapters plus one appendix. The first part of the volume focuses on the approximation of elliptic and mixed PDEs, beginning with fundamental results on well-posed weak formulations and their approximation by the Galerkin method. The material covered includes key results such as the BNB theorem based on inf-sup conditions, Céa's and Strang's lemmas, and the duality argument by Aubin and Nitsche. Important implementation aspects regarding quadratures, linear algebra, and assembling are also covered. The remainder of Volume II focuses on PDEs where a coercivity property is available. It investigates conforming and nonconforming approximation techniques (Galerkin, boundary penalty, Crouzeix—Raviart, discontinuous Galerkin, hybrid high-order methods). These techniques are applied to elliptic PDEs (diffusion, elasticity, the Helmholtz problem, Maxwell's equations), eigenvalue problems for elliptic PDEs, and PDEs in mixed form (Darcy and Stokes flows). Finally, the appendix addresses fundamental results on the surjectivity, bijectivity, and coercivity of linear operators in Banach spaces.
Authors and Affiliations
About the authors
Jean-Luc Guermond is Professor of Mathematics at Texas A&M University where he also holds an Exxon Mobile Chair in Computational Science. His current research interests are in numerical analysis, applied mathematics, and scientific computing. He has co-authored two books and over 170 research papers in peer-reviewed journals.
Bibliographic Information
Book Title: Finite Elements II
Book Subtitle: Galerkin Approximation, Elliptic and Mixed PDEs
Authors: Alexandre Ern, Jean-Luc Guermond
Series Title: Texts in Applied Mathematics
DOI: https://doi.org/10.1007/978-3-030-56923-5
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer Nature Switzerland AG 2021
Hardcover ISBN: 978-3-030-56922-8Published: 23 April 2021
Softcover ISBN: 978-3-030-56924-2Published: 23 April 2022
eBook ISBN: 978-3-030-56923-5Published: 22 April 2021
Series ISSN: 0939-2475
Series E-ISSN: 2196-9949
Edition Number: 1
Number of Pages: IX, 492
Number of Illustrations: 27 b/w illustrations, 1 illustrations in colour
Topics: Analysis, Functional Analysis