Finite elements. II, Galerkin approximation, elliptic and mixed PDEs / Alexandre Ern, Jean-Luc Guermond.
Ern, Alexandre, 1967- author.; Guermond, Jean-Luc, author.
2021
QA377
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Title Finite elements. II, Galerkin approximation, elliptic and mixed PDEs / Alexandre Ern, Jean-Luc Guermond.
Author Ern, Alexandre, 1967- author.
ISBN 9783030569235 (electronic bk.)
3030569233 (electronic bk.)
9783030569228 (print)
3030569225
DOI https://doi.org/10.1007/978-3-030-56923-5
Published Cham, Switzerland : Springer, [2021]
Language English
Description 1 online resource (ix, 492 pages) : illustrations (some color)
Other Standard Identifiers 10.1007/978-3-030-56923-5 doi
Call Number QA377
Dewey Decimal Classification 518/.25
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.
Bibliography, etc. Note Includes bibliographical references and index.
Access Note Access limited to authorized users.
Source of Description Online resource; title from PDF title page (SpringerLink, viewed April 27, 2021).
Added Author Guermond, Jean-Luc, author.
Series Texts in applied mathematics ; 73. 0939-2475
Available in Other Form Finite elements. II, Galerkin approximation, elliptic and mixed PDEs.
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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 equations
Potential and flux recovery
Stokes equations: Basic ideas
Stokes equations: Stable Pairs (I)
Stokes equations: Stable pairs (II)
Appendices
Bijective operators in Banach spaces.

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