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.
Author(s): Alexandre Ern, Jean-Luc Guermond
Series: Texts in Applied Mathematics 72
Edition: 1
Publisher: Springer Nature Switzerland
Year: 2021
Language: English
Pages: 325
City: Cham
Tags: Sobolev Spaces, Weak Derivatives, Finite Elements, Meshes
Preface
Contents
Part I Elements of functional analysis
1 Lebesgue spaces
1.1 Heuristic motivation
1.2 Lebesgue measure
1.3 Lebesgue integral
1.4 Lebesgue spaces
1.4.1 Lebesgue space L1(D)
1.4.2 Lebesgue spaces Lp(D) and Linfty(D)
1.4.3 Duality
1.4.4 Multivariate functions
2 Weak derivatives and Sobolev spaces
2.1 Differentiation
2.1.1 Lebesgue points
2.1.2 Weak derivatives
2.2 Sobolev spaces
2.2.1 Integer-order spaces
2.2.2 Fractional-order spaces
2.3 Key properties: density and embedding
2.3.1 Density of smooth functions
2.3.2 Embedding
3 Traces and Poincaré inequalities
3.1 Lipschitz sets and domains
3.2 Traces as functions at the boundary
3.2.1 The spaces Ws,p0(D), Ws,p(D) and their traces
3.2.2 The spaces widetildeWs,p(D)
3.3 Poincaré–Steklov inequalities
4 Distributions and duality in Sobolev spaces
4.1 Distributions
4.2 Negative-order Sobolev spaces
4.3 Normal and tangential traces
Part II Introduction to finite elements
5 Main ideas and definitions
5.1 Introductory example
5.2 Finite element as a triple
5.3 Interpolation: finite element as a quadruple
5.4 Basic examples
5.4.1 Lagrange (nodal) finite elements
5.4.2 Modal finite elements
5.5 The Lebesgue constant
6 One-dimensional finite elements and tensorization
6.1 Legendre and Jacobi polynomials
6.2 One-dimensional Gauss quadrature
6.3 One-dimensional finite elements
6.3.1 Lagrange (nodal) finite elements
6.3.2 Modal finite elements
6.3.3 Canonical hybrid finite element
6.3.4 Hierarchical bases
6.3.5 High-order Lagrange elements
6.4 Multidimensional tensor-product elements
6.4.1 The polynomial space mathbbQk,d
6.4.2 Tensor-product construction of finite elements
6.4.3 Serendipity finite elements
7 Simplicial finite elements
7.1 Simplices
7.2 Barycentric coordinates, geometric mappings
7.3 The polynomial space mathbbPk,d
7.4 Lagrange (nodal) finite elements
7.5 Crouzeix–Raviart finite element
7.6 Canonical hybrid finite element
Part III Finite element interpolation
8 Meshes
8.1 The geometric mapping
8.2 Main definitions related to meshes
8.3 Data structure
8.4 Mesh generation
8.4.1 Two-dimensional case
8.4.2 Three-dimensional case
9 Finite element generation
9.1 Main ideas
9.2 Differential calculus and geometry
9.2.1 Transformation of differential operators
9.2.2 Normal and tangent vectors
10 Mesh orientation
10.1 How to orient a mesh
10.2 Generation-compatible orientation
10.3 Increasing vertex-index enumeration
10.4 Simplicial meshes
10.5 Quadrangular and hexahedral meshes
11 Local interpolation on affine meshes
11.1 Shape-regularity for affine meshes
11.2 Transformation of Sobolev seminorms
11.3 Bramble–Hilbert lemmas
11.4 Local finite element interpolation
11.5 Some examples
11.5.1 Lagrange elements
11.5.2 Modal elements
11.5.3 L2-orthogonal projection
12 Local inverse and functional inequalities
12.1 Inverse inequalities in cells
12.2 Inverse inequalities on faces
12.3 Functional inequalities in meshes
12.3.1 Poincaré–Steklov inequality in cells
12.3.2 Multiplicative trace inequality
13 Local interpolation on nonaffine meshes
13.1 Introductory example on curved simplices
13.2 A perturbation theory
13.2.1 Setting and notation
13.2.2 Bounds on the derivatives of T and T-1
13.3 Interpolation error on nonaffine meshes
13.3.1 Transformation of Sobolev norms
13.3.2 Bramble–Hilbert lemmas in mathbbQk,d
13.3.3 Interpolation error estimates
13.4 Curved simplices
13.5 mathbbQ1-quadrangles
13.6 mathbbQ2-curved quadrangles
14 H(div) finite elements
14.1 The lowest-order case
14.2 The polynomial space mathbbRT- .4 k,d
14.3 Simplicial Raviart–Thomas elements
14.4 Generation of Raviart–Thomas elements
14.5 Other H(div) finite elements
14.5.1 Brezzi–Douglas–Marini elements
14.5.2 Cartesian Raviart–Thomas elements
15 H(curl) finite elements
15.1 The lowest-order case
15.2 The polynomial space mathbbN-.4k,d
15.3 Simplicial Nédélec elements
15.3.1 Two-dimensional case
15.3.2 Three-dimensional case
15.4 Generation of Nédélec elements
15.5 Other H(curl) finite elements
15.5.1 Nédélec elements of the second kind
15.5.2 Cartesian Nédélec elements
16 Local interpolation in H(div) and H(curl) (I)
16.1 Local interpolation in H(div)
16.1.1 Extending the dofs
16.1.2 Commuting and approximation properties
16.2 Local interpolation in H(curl)
16.2.1 Extending the dofs
16.2.2 Commuting and approximation properties
16.3 The de Rham complex
17 Local interpolation in H(div) and H(curl) (II)
17.1 Face-to-cell lifting operator
17.2 Local interpolation in H(div) using liftings
17.3 Local interpolation in H(curl) using liftings
Part IV Finite element spaces
18 From broken to conforming spaces
18.1 Broken spaces and jumps
18.1.1 Broken Sobolev spaces and jumps
18.1.2 Broken finite element spaces
18.2 Conforming finite element subspaces
18.2.1 Membership in H1
18.2.2 Membership in H(curl) and H(div)
18.2.3 Unified notation for conforming subspaces
18.3 L1-stable local interpolation
18.4 Broken L2-orthogonal projection
19 Main properties of the conforming subspaces
19.1 Global shape functions and dofs
19.2 Examples
19.2.1 H1-conforming subspace Pkg(mathcalTh)
19.2.2 H(curl)-conforming subspace Pkc(mathcalTh)
19.2.3 H(div)-conforming subspace Pkd(mathcalTh)
19.3 Global interpolation operators
19.4 Subspaces with zero boundary trace
20 Face gluing
20.1 The two gluing assumptions (Lagrange)
20.2 Verification of the assumptions (Lagrange)
20.2.1 Face unisolvence
20.2.2 The space PK,F
20.2.3 Face matching
20.3 Generalization of the two gluing assumptions
20.4 Verification of the two gluing assumptions
20.4.1 Raviart–Thomas elements
20.4.2 Nédélec elements
20.4.3 Canonical hybrid elements
21 Construction of the connectivity classes
21.1 Connectivity classes
21.1.1 Geometric entities and macroelements
21.1.2 The two key assumptions
21.1.3 Connectivity classes as equivalence classes
21.2 Verification of the assumptions
21.2.1 Lagrange and canonical hybrid elements
21.2.2 Nédélec elements
21.2.3 Raviart–Thomas elements
21.3 Practical construction
21.3.1 Enumeration of the geometric entities in K"0362K
21.3.2 Example of a construction of χlr and j _ _dof
22 Quasi-interpolation and best approximation
22.1 Discrete setting
22.2 Averaging operator
22.3 Quasi-interpolation operator
22.4 Quasi-interpolation with zero trace
22.4.1 Averaging operator revisited
22.4.2 Quasi-interpolation operator revisited
22.5 Conforming L2-orthogonal projections
23 Commuting quasi-interpolation
23.1 Smoothing by mollification
23.2 Mesh-dependent mollification
23.3 L1-stable commuting projection
23.3.1 First step: the operator calIh°mathcalKδ
23.3.2 Second step: the operator Jh °calIh °mathcalKδ
23.3.3 Main results
23.4 Mollification with extension by zero
A Banach and Hilbert spaces
Appendix B Differential calculus
References
Index
