hp-Finite Element Methods for Singular Perturbations

6y2b766vkg3uemaw.pdf......Page 0
Sets, balls, sectors, neighborhoods......Page 11
Norms, difeerential operators, standard function spaces......Page 12
Description of the boundary and corner layer......Page 13
Meshes and finite element approximation......Page 14
1.2 Problem class and assumptions......Page 15
1.3 Principal results......Page 18
1.4.1 Elliptic problems in non-smooth domains......Page 21
1.4.2 Regularity in terms of asymptotic expansions......Page 23
1.4.3 {it hp} finite element methods......Page 29
1.4.4 Numerical methods for singular perturbations......Page 30
1.5 Outline of the book......Page 33
2.1 Setting and Introduction......Page 35
2.2.1 Regularity in one dimension......Page 36
2.2.2 {it hp}-FEM in one dimension......Page 37
2.2.3 Numerical examples......Page 42
2.3 Regularity: the two-dimensional case......Page 43
2.4.1 {it hp}-meshes and spaces......Page 50
2.4.2 The minimal {it hp}-mesh......Page 51
2.4.3 {it hp}-FEM......Page 55
2.5 Numerical Examples......Page 57
2.5.1 The classical L-shaped domain......Page 58
2.5.2 Robustness with respect to mesh distortion......Page 61
2.5.3 Examples with singular right-hand side......Page 62
2.6 {it h}-FEM approximation......Page 66
2.6.1 Approximation on Shishkin meshes in one dimension......Page 67
2.6.2 {it h}-FEM meshes......Page 69
2.6.3 {it h}-FEM boundary layer meshes......Page 74
3.1.1 General overview of Chapter 3......Page 85
3.1.2 Outline of Chapter 3......Page 87
3.1.3 Robust exponential convergence: key ingredients of proof......Page 88
3.2.1 Notation and properties of polynomials......Page 99
3.2.2 Approximation of analytic functions: intervals and squares......Page 102
3.2.3 Approximation of analytic functions on triangles......Page 105
3.2.4 The projector $H^{infty}_{p}$......Page 115
3.2.5 Anisotropic projection operators: $H^{1,infty}_{p}$......Page 117
3.2.6 An optimal error estimate for an $H^{1}$-projector......Page 121
3.3 Admissible boundary layer meshes and .nite element spaces......Page 123
3.3.1 {it hp}-meshes for the approximation of boundary and corner layers......Page 126
3.3.2 Patchwise structured meshes......Page 128
3.3.3 The {it p}-version boundary layer and corner layer patches......Page 130
3.3.4 Boundary layer mesh generation via mesh patches......Page 132
3.3.5 Properties of the pull-backs to the patches......Page 134
3.4.1 Regularity on the reference element......Page 135
3.4.2 Approximation on minimal meshes......Page 139
4.1.1 Motivation......Page 151
4.2 The spaces $H^{m,l}_{beta,varepsilon}$ and $mathcal{B}^{l}_{beta,varepsilon}$ in a Sector......Page 156
4.2.1 Properties of the spaces $H^{m,l}_{beta,varepsilon}(Omega)$......Page 160
4.2.2 Properties of the countably normed spaces $mathcal{B}^{l}_{beta,varepsilon}$......Page 164
4.3 Local changes of variables for analytic functions......Page 175
5.1.1 Motivation......Page 179
5.1.2 Outline of Chapter 5......Page 182
5.2 Analytic regularity results of Babu{s}ka and Guo......Page 183
5.3 Analytic regularity: Dirichlet problems......Page 186
5.3.1 Analytic regularity in sectors......Page 187
5.3.2 Regularity in curvilinear polygons......Page 194
5.4.1 Neumann and Robin corners......Page 198
5.4.2 Mixed corners......Page 205
5.4.3 Transmission problems......Page 206
5.5 Local regularity......Page 207
5.5.1 Preliminaries: local $H^{2}$-regularity......Page 210
5.5.2 Interior regularity: Proof of Proposition 5.5.1......Page 212
5.5.3 Regularity at the boundary: Proof of Proposition 5.5.2......Page 218
5.5.4 Regularity of transmission problems: Proof of Prop. 5.5.4......Page 225
5.5.5 Regularity of Neumann problems: Proof of Prop. 5.5.3......Page 234
6.1.1 Motivation......Page 235
6.1.2 Outline of Chapter 6......Page 238
6.2 The exponentially weighted spaces ${H}^{m,l}_{β,ε,α}$ and $mathcal{B}^{l}_{β,ε,α}$ in sectors......Page 239
6.3 Change of variables: from polar to Cartesian coordinates......Page 243
6.4.1 Transmission problem: problem formulation......Page 246
6.4.2 Transmission problem in exponentially weighted spaces......Page 251
6.4.3 Analytic regularity in exponentially weighted spaces......Page 254
6.4.4 Analytic regularity for a special transmission problem......Page 256
7.1.1 Motivation......Page 263
7.1.2 Outline of Chapter 7......Page 270
7.2 Regularity of the outer expansion......Page 271
7.3.1 Definition and properties of the boundary layer expansion......Page 275
7.3.2 Proof of Theorem 7.3.3......Page 278
7.4.1 Notation and main result......Page 291
7.4.2 Proof of Theorem 7.4.5: smooth and boundary layer parts......Page 296
7.4.3 Proof of Theorem 7.4.5: corner layer and remainder......Page 298
A.1.1 Transformations of elliptic equations......Page 304
A.1.2 Leibniz formulas......Page 305
A.1.3 Hardy inequalities......Page 308
A.2.1 Problem formulation and notation......Page 309
A.2.2 Proof of Proposition A.2.1......Page 310
A.3 Stability properties of the Gauss-Lobatto interpolant......Page 316
A.4 $L^{infty}$ projectors......Page 317