Finite elements

by Ern, Alexandre

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Title Statements

Full title :Finite elements. / Alexandre Ern, Jean-Luc Guermond.

Summary

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.

Edition Statements

Publication Date:2021

Publication Statements

Publisher(s):Springer
Place of Publication:Cham, Switzerland
Language:eng
Publication Date:2021

The Physical Object

Description:1 online resource (ix, 492 pages) : : illustrations (some color)

Edition Identifiers

ISBN:9783030569235, 3030569233
DOI:10.1007/978-3-030-56923-5
oclc:on1247885582
LC Classification:QA377
Dewey Decimal:518/.25

Part V: Weak formulations and well-posedness -- Weak formulation of model problems -- Main results on well-posedness -- Part VI: Galerkin approximation -- Basic error analysis -- Error analysis with variational crimes -- Linear algebra -- Sparse matrices -- Quadratures -- Part VII: Elliptic PDEs: conforming approximation -- Scalar second-order elliptic PDEs -- H1-conforming approximation (I) -- H1-conforming approximation (II) -- A posteriori error analysis -- The Helmholtz problem -- Part VIII: Elliptic PDEs: nonconforming approximation -- Crouzeix-Raviart approximation -- Nitsche's boundary penalty method -- Discontinuous Galerkin -- Hybrid high-order methods -- Contrasted diffusivity (I) -- Contrasted diffusivity (II) -- Part IX: Vector-valued elliptic PDEs -- Linear elasticity -- Maxwell's equations: H(curl)-approximation -- Maxwell's equations: control on the divergence -- Maxwell's equations: further topics -- Part X: Eigenvalue problems -- Symmetric elliptic eigenvalue problems -- Symmetric operators, conforming approximation -- Nonsymmetric problems -- Part XI: PDEs in mixed form -- Well-posedness for PDEs in mixed form -- Mixed finite element approximation -- Darcy's equa…

Subjects

Anàlisi funcional Analyse fonctionnelle Differential equations, Partial -- Nume… Equacions en derivades parcials Équations aux dérivées partielles -- So… Finite element method Functional analysis Galerkin methods Méthode de Galerkin Méthode des éléments finis Mètode dels elements finits Mètodes de Galerkin