Finite Elements II 1st Edition by Alexandre Ern, Jean Luc Guermond ISBN 3030569225 978-3030569228

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ISBN 10: 3030569225

ISBN 13: 978-3030569228

Author: Alexandre Ern, Jean-Luc Guermond

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.

Table of contents:

1. Weak formulation of model problems
2. Main results on well-posedness
3. Basic error analysis
4. Error analysis with variational crimes
5. Linear algebra
6. Sparse matrices
7. Quadratures
8. Scalar second-order elliptic PDEs
9. H1mathbb{H}^1-conforming approximation (I)
10. H1mathbb{H}^1-conforming approximation (II)
11. A posteriori error analysis
12. The Helmholtz problem
13. Crouzeix–Raviart approximation
14. Nitsche’s boundary penalty method
15. Discontinuous Galerkin
16. Hybrid high-order method
17. Contrasted diffusivity (I)
18. Contrasted diffusivity (II)
19. Linear elasticity
20. Maxwell’s equations: H(curl)-approximation
21. Maxwell’s equations: control on the divergence
22. Maxwell’s equations: further topics
23. Symmetric elliptic eigenvalue problems
24. Symmetric operators, conforming approximation
25. Nonsymmetric problems
26. Well-posedness for PDEs in mixed form
27. Mixed finite element approximation
28. Darcy’s equations
29. Potential and flux recovery
30. Stokes equations: Basic ideas
31. Stokes equations: Stable pairs (I)
32. Stokes equations: Stable pairs (II)

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Tags: Alexandre Ern, Jean Luc Guermond, Finite Elements