BEM-based Finite Element Approaches on Polytopal Meshes

Preface......Page 7
Contents......Page 9
 Function Spaces......Page 12
 Polytopal Mesh......Page 13
 Approximation Spaces, Interpolation and Projection......Page 14
 Boundary Integral Equations......Page 15
 Miscellaneous......Page 16
 1.1 Overview......Page 17
 1.2 Outline......Page 21
 1.3 Mathematical Preliminaries......Page 22
  1.3.1 Function Spaces and Trace Operators......Page 23
   1.3.1.1 Sobolev Spaces......Page 25
   1.3.1.2 Sobolev Spaces on the Boundary......Page 26
   1.3.1.3 Properties of Sobolev Spaces......Page 28
  1.3.2 Galerkin Formulations......Page 29
 2.1 Preliminaries......Page 33
 2.2 Polygonal and Polyhedral Meshes......Page 34
  2.2.1 Mesh Regularity and Properties in 2D......Page 36
  2.2.2 Mesh Regularity and Properties in 3D......Page 40
  2.2.3 Mesh Refinement......Page 47
 2.3 Trefftz-Like Basis Functions......Page 49
  2.3.1 Node and Edge Basis Functions......Page 50
  2.3.2 Element Basis Functions......Page 54
  2.3.3 Final Approximation Space......Page 56
  2.3.4 Simple Generalization to 3D......Page 57
 2.4 Interpolation Operators......Page 59
 2.5 Galerkin Formulation and Convergence Estimates......Page 69
 2.6 Numerical Examples......Page 74
 3.1 Preliminaries......Page 80
 3.2 Trace Inequality and Best Approximation......Page 83
  3.3.1 Clément-Type Interpolation......Page 87
  3.3.2 Scott–Zhang-Type Interpolation......Page 90
  3.4.1 Characterisation of Anisotropy and Regularity......Page 92
  3.4.2 Approximation Space......Page 101
  3.4.3 Anisotropic Trace Inequality and Best Approximation......Page 102
  3.4.4 Quasi-Interpolation of Anisotropic Non-smoothFunctions......Page 105
   3.4.4.1 Clément-Type Interpolation......Page 106
   3.4.4.2 Scott–Zhang-Type Interpolation......Page 109
  3.4.5 Interpolation of Anisotropic Smooth Functions......Page 110
  3.4.6 Numerical Assessment of Anisotropic Meshes......Page 113
 4.1 Preliminaries......Page 121
 4.2 Boundary Integral Formulations......Page 122
  4.2.1 Direct Approach for Dirichlet Problem......Page 125
  4.2.2 Indirect Approach for Dirichlet Problem......Page 126
  4.2.3 Direct Approach for Neumann Problem......Page 127
 4.3 Boundary Element Method......Page 129
  4.3.1 Dirichlet Problem......Page 130
  4.3.2 Neumann Problem......Page 135
  4.4.1 Domains with Smooth Boundary......Page 138
  4.4.2 Domains with Corners......Page 140
  4.5.1 Incorporation of Local Solvers and Quadratureon Polytopes......Page 144
  4.5.2 Numerical Examples and Comparison......Page 149
   4.5.2.1 Interpolation with Local Nyström Solver......Page 150
   4.5.2.2 Comparison of Local Solvers for L-Domain......Page 151
 5.1 Preliminaries......Page 155
 5.2 Residual Based Error Estimator......Page 159
  5.2.1 Reliability......Page 162
  5.2.2 Efficiency......Page 163
  5.2.3 Numerical Experiments......Page 166
 5.3 Goal-Oriented Error Estimation......Page 171
  5.3.1 DWR Method for Linear Goal Functionals......Page 172
  5.3.3 Approximation of the Primal and Dual Solution......Page 175
   5.3.3.2 Dual Solution Exploiting Local Post-Processing......Page 177
   5.3.4.1 The Classical Way of Localization......Page 179
   5.3.4.2 A Variational Error Estimator with PU Localization......Page 181
  5.3.5 Numerical Tests......Page 182
 6.1 Mixed Formulations Treated by Means of BEM-Based FEM......Page 191
  6.1.1 Mixed Formulation......Page 192
  6.1.2 H(div)-Conforming Approximation Space......Page 194
  6.1.3 Approximation of Mixed Formulation......Page 199
  6.1.4 Realization and Numerical Examples......Page 202
   6.1.4.1 Computational Realization......Page 203
   6.1.4.2 Numerical Examples......Page 205
  6.2.1 Generalization for Diffusion Problem......Page 209
  6.2.2 Application to Convection-Diffusion-Reaction Problem......Page 211
  6.2.3 Realization of the Basis Functions......Page 214
  6.2.4 Fully Discrete Galerkin Formulation......Page 219
  6.2.5 Numerical Experiments: Diffusion Problem......Page 221
  6.2.6 Numerical Experiments: Convection-Diffusion Problem......Page 225
References......Page 237
Index......Page 246