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E-raamat: Finite Elements I: Approximation and Interpolation

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  • Formaat: PDF+DRM
  • Sari: Texts in Applied Mathematics 72
  • Ilmumisaeg: 18-Feb-2021
  • Kirjastus: Springer Nature Switzerland AG
  • Keel: eng
  • ISBN-13: 9783030563417
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Finite Elements I: Approximation and InterpolationSuurem pilt
  • Formaat: PDF+DRM
  • Sari: Texts in Applied Mathematics 72
  • Ilmumisaeg: 18-Feb-2021
  • Kirjastus: Springer Nature Switzerland AG
  • Keel: eng
  • ISBN-13: 9783030563417
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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 finite elements 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.
Part I Elements of functional analysis
1 Lebesgue spaces
3(12)
1.1 Heuristic motivation
3(1)
1.2 Lebesgue measure
4(3)
1.3 Lebesgue integral
7(1)
1.4 Lebesgue spaces
8(7)
2 Weak derivatives and Sobolev spaces
15(12)
2.1 Differentiation
15(3)
2.2 Sobolev spaces
18(3)
2.3 Key properties: density and embedding
21(6)
3 Traces and Poincare inequalities
27(12)
3.1 Lipschitz sets and domains
27(3)
3.2 Traces as functions at the boundary
30(4)
3.3 Poincare--Steklov inequalities
34(5)
4 Distributions and duality in Sobolev spaces
39(10)
4.1 Distributions
39(2)
4.2 Negative-order Sobolev spaces
41(2)
4.3 Normal and tangential traces
43(6)
Part II Introduction to finite elements
5 Main ideas and definitions
49(10)
5.1 Introductory example
49(1)
5.2 Finite element as a triple
50(2)
5.3 Interpolation: finite element as a quadruple
52(1)
5.4 Basic examples
53(2)
5.5 The Lebesgue constant
55(4)
6 One-dimensional finite elements and tensorization
59(16)
6.1 Legendre and Jacobi polynomials
59(2)
6.2 One-dimensional Gauss quadrature
61(3)
6.3 One-dimensional finite elements
64(5)
6.4 Multidimensional tensor-product elements
69(6)
7 Simplicial finite elements
75(14)
7.1 Simplices
75(1)
7.2 Barycentric coordinates, geometric mappings
76(2)
7.3 The polynomial space Pk,d
78(1)
7.4 Lagrange (nodal) finite elements
78(4)
7.5 Crouzeix-Raviart finite element
82(1)
7.6 Canonical hybrid finite element
83(6)
Part III Finite element interpolation
8 Meshes
89(12)
8.1 The geometric mapping
89(2)
8.2 Main definitions related to meshes
91(4)
8.3 Data structure
95(2)
8.4 Mesh generation
97(4)
9 Finite element generation
101(10)
9.1 Main ideas
101(3)
9.2 Differential calculus and geometry
104(7)
10 Mesh orientation
111(12)
10.1 How to orient a mesh
111(2)
10.2 Generation-compatible orientation
113(2)
10.3 Increasing vertex-index enumeration
115(1)
10.4 Simplicial meshes
116(2)
10.5 Quadrangular and hexahedral meshes
118(5)
11 Local interpolation on affine meshes
123(14)
11.1 Shape-regularity for affine meshes
123(3)
11.2 Transformation of Sobolev seminorms
126(1)
11.3 Bramble-Hilbert lemmas
127(2)
11.4 Local finite element interpolation
129(3)
11.5 Some examples
132(5)
12 Local inverse and functional inequalities
137(10)
12.1 Inverse inequalities in cells
137(3)
12.2 Inverse inequalities on faces
140(2)
12.3 Functional inequalities in meshes
142(5)
13 Local interpolation on nonaffine meshes
147(14)
13.1 Introductory example on curved simplices
147(1)
13.2 A perturbation theory
148(3)
13.3 Interpolation error on nonaffine meshes
151(4)
13.4 Curved simplices
155(1)
13.5 Q1-quadrangles
156(2)
13.6 Q2-curved quadrangles
158(3)
14 H(div) finite elements
161(12)
14.1 The lowest-order case
161(2)
14.2 The polynomial space RTk,d
163(1)
14.3 Simplicial Raviart-Thomas elements
164(3)
14.4 Generation of Raviart-Thomas elements
167(2)
14.5 Other H(div) finite elements
169(4)
15 //(curl) finite elements
173(14)
15.1 The lowest-order case
173(3)
15.2 The polynomial space Nk,d
176(1)
15.3 Simplicial Ncdelec elements
177(5)
15.4 Generation of Nedelec elements
182(2)
15.5 Other H(curl) finite elements
184(3)
16 Local interpolation in H(div) and H(curl) (I)
187(12)
16.1 Local interpolation in H(div)
187(4)
16.2 Local interpolation in H(curl)
191(5)
16.3 The de Rham complex
196(3)
17 Local interpolation in H(div) and H(curl) (II)
199(16)
17.1 Face-to-cell lifting operator
199(3)
17.2 Local interpolation in H(div) using liftings
202(4)
17.3 Local interpolation in H(curl) using liftings
206(9)
Part IV Finite element spaces
18 From broken to conforming spaces
215(14)
18.1 Broken spaces and jumps
215(3)
18.2 Conforming finite element subspaces
218(4)
18.3 L1'-stable local interpolation
222(3)
18.4 Broken L2-orthogonal projection
225(4)
19 Main properties of the conforming subspaces
229(14)
19.1 Global shape functions and dofs
229(3)
19.2 Examples
232(3)
19.3 Global interpolation operators
235(4)
19.4 Subspaces with zero boundary trace
239(4)
20 Face gluing
243(14)
20.1 The two gluing assumptions (Lagrange)
243(2)
20.2 Verification of the assumptions (Lagrange)
245(4)
20.3 Generalization of the two gluing assumptions
249(2)
20.4 Verification of the two gluing assumptions
251(6)
21 Construction of the connectivity classes
257(16)
21.1 Connectivity classes
257(7)
21.2 Verification of the assumptions
264(1)
21.3 Practical construction
265(8)
22 Quasi-interpolation and best approximation
273(14)
22.1 Discrete setting
273(2)
22.2 Averaging operator
275(2)
22.3 Quasi-interpolation operator
277(3)
22.4 Quasi-interpolation with zero trace
280(3)
22.5 Conforming L2-orthogonal projections
283(4)
23 Commuting quasi-interpolation
287(16)
23.1 Smoothing by mollification
287(3)
23.2 Mesh-dependent mollification
290(2)
23.3 L1 - stable commuting projection
292(6)
23.4 Mollification with extension by zero
298(5)
Appendices
A Banach and Hilbert spaces
303(6)
A.1 Banach spaces
303(1)
A.2 Bounded linear maps and duality
304(1)
A.3 Hilbert spaces
305(1)
A.4 Compact operators
306(1)
A.5 Interpolation between Banach spaces
307(2)
B Differential calculus
309(4)
B.1 Frechet derivative
309(1)
B.2 Vector and matrix representation
310(3)
References 313(10)
Index 323
Alexandre Ern is Senior Researcher at Ecole des Ponts and INRIA in Paris, and he is also Associate Professor of Numerical Analysis at Ecole Polytechnique, Paris. His research deals with the devising and analysis of finite element methods and a posteriori error estimates and adaptivity with applications to fluid and solid mechanics and porous media flows. Alexandre Ern has co-authored three books and over 150 papers in peer-reviewed journals. He has supervised about 20 PhD students and 10 post-doctoral fellows, and he has ongoing collaborations with several industrial partners.  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.
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