دانلود کتاب Finite Elements I Approximation and Interpolation

فهرست مطالب :

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