Splines and PDEs: From approximation theory to numerical linear algebra

Preface......Page 6
Acknowledgments......Page 8
Contents......Page 9
 1.1 Introduction......Page 10
  1.2.1 Definition and Basic Properties......Page 12
  1.2.2 Dual Polynomials......Page 16
  1.2.3 Local Marsden Identity and Linear Independence......Page 17
  1.2.4 Smoothness, Differentiation and Integration......Page 19
  1.3.1 The Spline Space Sp,ξ and Some Spline Properties......Page 23
  1.3.2 The Piecewise Polynomial Space Srp(Δ)......Page 26
  1.3.3 B-Spline Representation of Polynomials......Page 29
  1.3.4 B-Spline Representation of Splines......Page 32
  1.3.5 Knot Insertion......Page 34
  1.3.6 Condition Number......Page 36
  1.4.1 Main Properties......Page 39
  1.4.2 Inner Products......Page 43
  1.4.3 Uniform Knot Insertion......Page 44
 1.5 Spline Approximation......Page 46
  1.5.1 Preliminaries......Page 47
  1.5.2 Spline Quasi-Interpolation......Page 48
  1.5.3 Approximation Power of Splines......Page 54
   1.5.3.1 A Spline Quasi-Interpolant......Page 55
   1.5.3.2 Distance to a Function......Page 58
  1.6.1 Hierarchical B-Splines......Page 60
  1.6.2 Truncated Hierarchical B-Splines......Page 65
  1.6.3 Quasi-Interpolation in Hierarchical Spaces......Page 71
  1.7.1 Tensor-Product B-Splines......Page 77
  1.7.2 Local Refinement......Page 80
 References......Page 83
 2.1 Introduction......Page 86
 2.2 BPX Preconditioning for Isogeometric Analysis......Page 89
  2.2.1 B-Spline Discretizations......Page 91
  2.2.2 Additive Multilevel Preconditioners......Page 93
  2.2.3 Realization of the BPX Preconditioner......Page 97
  2.3.1 An Abstract Operator Equation......Page 102
  2.3.2 Elliptic Boundary Value Problems......Page 104
  2.3.3 Saddle Point Problems Involving Boundary Conditions......Page 105
  2.3.4 Parabolic Boundary Value Problems......Page 109
  2.3.5 PDE-Constrained Control Problems: Distributed Control......Page 112
  2.3.6 PDE-Constrained Control Problems: Dirichlet Boundary Control......Page 114
  2.3.7 PDE-Constrained Control Problems: Parabolic PDEs......Page 115
  2.4.1 Basic Properties......Page 116
  2.4.2 Norm Equivalences and Riesz Maps......Page 119
  2.4.4 Multiscale Decomposition of Function Spaces......Page 121
  2.5.1 Elliptic Boundary Value Problems......Page 137
  2.5.2 Saddle Point Problems Involving Boundary Conditions......Page 139
  2.5.3 Control Problems: Distributed Control......Page 141
  2.5.4 Control Problems: Dirichlet Boundary Control......Page 146
 2.6 Iterative Solution......Page 148
  2.6.1 Finite Systems on Uniform Grids......Page 149
  2.6.2 Adaptive Schemes......Page 156
 References......Page 165
 3.1 Introduction......Page 169
 3.2 The Theory of GLT Sequences: A Summary......Page 172
   3.3.1.1 Matrix-Norm Inequalities......Page 176
   3.3.1.2 GLT Preconditioning......Page 177
   3.3.1.3 Arrow-Shaped Sampling Matrices......Page 178
   3.3.2.1 FD Discretization of Diffusion Equations......Page 180
   3.3.2.2 FD Discretization of Convection-Diffusion-Reaction Equations......Page 183
   3.3.2.3 FD Discretization of Higher-Order Equations......Page 194
   3.3.2.4 Non-uniform FD Discretizations......Page 197
   3.3.3.1 FE Discretization of Convection-Diffusion-Reaction Equations......Page 202
   3.3.3.2 FE Discretization of a System of Equations......Page 208
  3.3.4 IgA Discretization of Differential Equations......Page 211
   3.3.4.1 B-Spline IgA Collocation Discretization of Convection-Diffusion-Reaction Equations......Page 212
   3.3.4.2 Galerkin B-Spline IgA Discretization of Convection-Diffusion-Reaction Equations......Page 225
   3.3.4.3 Galerkin B-Spline IgA Discretization of Second-Order Eigenvalue Problems......Page 236
 References......Page 241
 4.1 Introduction......Page 245
 4.2 Splines and NURBS: Definition and Properties......Page 246
  4.2.1 Univariate Splines......Page 247
  4.2.2 Multivariate Splines and NURBS......Page 251
  4.2.3 Splines Spaces with Local Tensor-Product Structure......Page 253
 4.3 Isogeometric Spaces: Definition......Page 256
  4.3.1 Isoparametric Spaces......Page 257
  4.3.2 De Rham Compatible Spaces......Page 258
  4.4.1 h-Refinement......Page 261
  4.4.2 p-Refinement and k-Refinement......Page 266
  4.4.3 Multipatch......Page 270
  4.4.4 Singular Parametrizations......Page 275
 4.5 Isogeometric Spaces: Spectral Properties......Page 276
  4.5.1 Spectrum and Dispersion Analysis......Page 277
  4.5.2 Eigenfunction Approximation......Page 288
  4.6.1 Formation of Isogeometric Matrices......Page 292
  4.6.2 Weighted Quadrature......Page 293
  4.6.3 Linear Solvers and Preconditioners......Page 299
  4.6.4 Matrix-Free Computationally-Efficient k-Refinement......Page 306
  4.7.1 Linear Elasticity......Page 312
  4.7.2 Steady Navier-Stokes Problem......Page 315
 References......Page 318