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＜電子ブック＞
Finite Elements I : Approximation and Interpolation / by Alexandre Ern, Jean-Luc Guermond
(Texts in Applied Mathematics. ISSN:21969949 ; 72)

版	1st ed. 2021.
出版者	(Cham : Springer International Publishing : Imprint: Springer)
出版年	2021
本文言語	英語
大きさ	XII, 325 p. 90 illus : online resource
著者標目	*Ern, Alexandre author Guermond, Jean-Luc author SpringerLink (Online service)
件　名	LCSH:Differential equations LCSH:Functional analysis FREE:Differential Equations FREE:Functional Analysis
一般注記	Part I: Elements of Functional Analysis -- Lebesgue spaces -- Weak derivatives and Sobolev spaces -- Traces and Poincaré Inequalities -- Duality in Sobolev spaces -- Part II: Introduction to Finite Elements -- Main ideas and definitions -- One-dimensional finite elements and tensorization -- Simplicial finite elements -- Part III: Finite element interpolation -- Meshes -- Finite element generation -- Mesh orientation -- Local interpolation on affine meshes -- Local inverse and functional inequalities -- Local interpolation on non-affine meshes -- H(div) finite elements -- H(curl) finite elements -- Local interpolation in H(div) and H(curl) (I) -- Local interpolation in H(div) and H(curl) (II) -- Part IV: Finite element spaces -- From broken to conforming spaces -- Main properties of the conforming spaces -- Face gluing -- Construction of the connectivity classes -- Quasi-interpolation and best approximation -- Commuting quasi-interpolation -- Appendices -- Banach and Hillbert spaces -- Differential calculus This book is the first 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 I is divided into 23 chapters plus two appendices on Banach and Hilbert spaces and on differential calculus. This volume focuses on the fundamental ideas regarding the construction of finiteelements and their approximation properties. It addresses the all-purpose Lagrange finite elements, but also vector-valued finite elements that are crucial to approximate the divergence and the curl operators. In addition, it also presents and analyzes quasi-interpolation operators and local commuting projections. The volume starts with four chapters on functional analysis, which are packed with examples and counterexamples to familiarize the reader with the basic facts on Lebesgue integration and weak derivatives. Volume I also reviews important implementation aspects when either developing or using a finite element toolbox, including the orientation of meshes and the enumeration of the degrees of freedom HTTP:URL=https://doi.org/10.1007/978-3-030-56341-7

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