Finite Elements I : Approximation and Interpolation 1

"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."-- Provided by publisher Preface 6 Contents 10 Part I Elements of functional analysis 14 1 Lebesgue spaces 15 1.1 Heuristic motivation 15 1.2 Lebesgue measure 16 1.3 Lebesgue integral 19 1.4 Lebesgue spaces 20 1.4.1 Lebesgue space L1(D) 20 1.4.2 Lebesgue spaces Lp(D) and Linfty(D) 22 1.4.3 Duality 23 1.4.4 Multivariate functions 24 2 Weak derivatives and Sobolev spaces 26 2.1 Differentiation 26 2.1.1 Lebesgue points 26 2.1.2 Weak derivatives 27 2.2 Sobolev spaces 29 2.2.1 Integer-order spaces 29 2.2.2 Fractional-order spaces 31 2.3 Key properties: density and embedding 32 2.3.1 Density of smooth functions 33 2.3.2 Embedding 33 3 Traces and Poincaré inequalities 38 3.1 Lipschitz sets and domains 38 3.2 Traces as functions at the boundary 41 3.2.1 The spaces Ws,p0(D), Ws,p(D) and their traces 41 3.2.2 The spaces widetildeWs,p(D) 43 3.3 Poincaré–Steklov inequalities 45 4 Distributions and duality in Sobolev spaces 49 4.1 Distributions 49 4.2 Negative-order Sobolev spaces 51 4.3 Normal and tangential traces 53 Part II Introduction to finite elements 56 5 Main ideas and definitions 57 5.1 Introductory example 57 5.2 Finite element as a triple 58 5.3 Interpolation: finite element as a quadruple 60 5.4 Basic examples 61 5.4.1 Lagrange (nodal) finite elements 61 5.4.2 Modal finite elements 62 5.5 The Lebesgue constant 63 6 One-dimensional finite elements and tensorization 67 6.1 Legendre and Jacobi polynomials 67 6.2 One-dimensional Gauss quadrature 69 6.3 One-dimensional finite elements 72 6.3.1 Lagrange (nodal) finite elements 72 6.3.2 Modal finite elements 73 6.3.3 Canonical hybrid finite element 74 6.3.4 Hierarchical bases 75 6.3.5 High-order Lagrange elements 76 6.4 Multidimensional tensor-product elements 77 6.4.1 The polynomial space mathbbQk,d 77 6.4.2 Tensor-product construction of finite elements 77 6.4.3 Serendipity finite elements 79 7 Simplicial finite elements 82 7.1 Simplices 82 7.2 Barycentric coordinates, geometric mappings 83 7.3 The polynomial space mathbbPk,d 85 7.4 Lagrange (nodal) finite elements 85 7.5 Crouzeix–Raviart finite element 89 7.6 Canonical hybrid finite element 90 Part III Finite element interpolation 94 8 Meshes 95 8.1 The geometric mapping 95 8.2 Main definitions related to meshes 97 8.3 Data structure 101 8.4 Mesh generation 103 8.4.1 Two-dimensional case 103 8.4.2 Three-dimensional case 104 9 Finite element generation 107 9.1 Main ideas 107 9.2 Differential calculus and geometry 110 9.2.1 Transformation of differential operators 110 9.2.2 Normal and tangent vectors 112 10 Mesh orientation 117 10.1 How to orient a mesh 117 10.2 Generation-compatible orientation 119 10.3 Increasing vertex-index enumeration 121 10.4 Simplicial meshes 122 10.5 Quadrangular and hexahedral meshes 124 11 Local interpolation on affine meshes 128 11.1 Shape-regularity for affine meshes 128 11.2 Transformation of Sobolev seminorms 131 11.3 Bramble–Hilbert lemmas 132 11.4 Local finite element interpolation 134 11.5 Some examples 137 11.5.1 Lagrange elements 137 11.5.2 Modal elements 137 11.5.3 L2-orthogonal projection 138 12 Local inverse and functional inequalities 142 12.1 Inverse inequalities in cells 142 12.2 Inverse inequalities on faces 145 12.3 Functional inequalities in meshes 147 12.3.1 Poincaré–Steklov inequality in cells 147 12.3.2 Multiplicative trace inequality 148 13 Local interpolation on nonaffine meshes 152 13.1 Introductory example on curved simplices 152 13.2 A perturbation theory 153 13.2.1 Setting and notation 153 13.2.2 Bounds on the derivatives of T and T-1 154 13.3 Interpolation error on nonaffine meshes 156 13.3.1 Transformation of Sobolev norms 156 13.3.2 Bramble–Hilbert lemmas in mathbbQk,d 158 13.3.3 Interpolation error estimates 158 13.4 Curved simplices 160 13.5 mathbbQ1-quadrangles 161 13.6 mathbbQ2-curved quadrangles 163 14 H(div) finite elements 166 14.1 The lowest-order case 166 14.2 The polynomial space mathbbRT- .4 k,d 168 14.3 Simplicial Raviart–Thomas elements 169 14.4 Generation of Raviart–Thomas elements 172 14.5 Other H(div) finite elements 174 14.5.1 Brezzi–Douglas–Marini elements 174 14.5.2 Cartesian Raviart–Thomas elements 175 15 H(curl) finite elements 178 15.1 The lowest-order case 178 15.2 The polynomial space mathbbN-.4k,d 181 15.3 Simplicial Nédélec elements 182 15.3.1 Two-dimensional case 183 15.3.2 Three-dimensional case 183 15.4 Generation of Nédélec elements 187 15.5 Other H(curl) finite elements 189 15.5.1 Nédélec elements of the second kind 189 15.5.2 Cartesian Nédélec elements 189 16 Local interpolation in H(div) and H(curl) (I) 192 16.1 Local interpolation in H(div) 192 16.1.1 Extending the dofs 192 16.1.2 Commuting and approximation properties 194 16.2 Local interpolation in H(curl) 196 16.2.1 Extending the dofs 196 16.2.2 Commuting and approximation properties 197 16.3 The de Rham complex 201 17 Local interpolation in H(div) and H(curl) (II) 204 17.1 Face-to-cell lifting operator 204 17.2 Local interpolation in H(div) using liftings 207 17.3 Local interpolation in H(curl) using liftings 211 Part IV Finite element spaces 217 18 From broken to conforming spaces 218 18.1 Broken spaces and jumps 218 18.1.1 Broken Sobolev spaces and jumps 218 18.1.2 Broken finite element spaces 220 18.2 Conforming finite element subspaces 221 18.2.1 Membership in H1 221 18.2.2 Membership in H(curl) and H(div) 222 18.2.3 Unified notation for conforming subspaces 223 18.3 L1-stable local interpolation 225 18.4 Broken L2-orthogonal projection 228 19 Main properties of the conforming subspaces 231 19.1 Global shape functions and dofs 231 19.2 Examples 234 19.2.1 H1-conforming subspace Pkg(mathcalTh) 234 19.2.2 H(curl)-conforming subspace Pkc(mathcalTh) 236 19.2.3 H(div)-conforming subspace Pkd(mathcalTh) 237 19.3 Global interpolation operators 237 19.4 Subspaces with zero boundary trace 241 20 Face gluing 245 20.1 The two gluing assumptions (Lagrange) 245 20.2 Verification of the assumptions (Lagrange) 247 20.2.1 Face unisolvence 247 20.2.2 The space PK,F 248 20.2.3 Face matching 248 20.3 Generalization of the two gluing assumptions 251 20.4 Verification of the two gluing assumptions 253 20.4.1 Raviart–Thomas elements 253 20.4.2 Nédélec elements 255 20.4.3 Canonical hybrid elements 256 21 Construction of the connectivity classes 259 21.1 Connectivity classes 259 21.1.1 Geometric entities and macroelements 260 21.1.2 The two key assumptions 261 21.1.3 Connectivity classes as equivalence classes 263 21.2 Verification of the assumptions 266 21.2.1 Lagrange and canonical hybrid elements 266 21.2.2 Nédélec elements 267 21.2.3 Raviart–Thomas elements 267 21.3 Practical construction 267 21.3.1 Enumeration of the geometric entities in K"0362K 267 21.3.2 Example of a construction of χlr and j _ _dof 268 22 Quasi-interpolation and best approximation 274 22.1 Discrete setting 274 22.2 Averaging operator 276 22.3 Quasi-interpolation operator 278 22.4 Quasi-interpolation with zero trace 281 22.4.1 Averaging operator revisited 281 22.4.2 Quasi-interpolation operator revisited 282 22.5 Conforming L2-orthogonal projections 284 23 Commuting quasi-interpolation 288 23.1 Smoothing by mollification 288 23.2 Mesh-dependent mollification 291 23.3 L1-stable commuting projection 293 23.3.1 First step: the operator calIh°mathcalKδ 293 23.3.2 Second step: the operator Jh °calIh °mathcalKδ 296 23.3.3 Main results 297 23.4 Mollification with extension by zero 299 A Banach and Hilbert spaces 303 Appendix B Differential calculus 309 References 312 Index 321 Part I: Elements of Functional Analysis -- Lebesgue spaces -- Weak derivatives and Sobolev spaces -- Traces and Poincaré Inequalities -- Duality in Sobolev spaces -- Part II: Introduction to Finite Elements -- Main ideas and definitions -- One-dimensional finite elements and tensorization -- Simplicial finite elements -- Part III: Finite element interpolation -- Meshes -- Finite element generation -- Mesh orientation -- Local interpolation on affine meshes -- Local inverse and functional inequalities -- Local interpolation on non-affine meshes -- H(div) finite elements -- H(curl) finite elements -- Local interpolation in H(div) and H(curl) (I) -- Local interpolation in H(div) and H(curl) (II) -- Part IV: Finite element spaces -- From broken to conforming spaces -- Main properties of the conforming spaces -- Face gluing -- Construction of the connectivity classes -- Quasi-interpolation and best approximation -- Commuting quasi-interpolation -- Appendices -- Banach and Hillbert spaces -- Differential calculus