Finite Elements I Approximation and Interpolation

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