The Hybrid High-Order Method for Polytonal Meshes: Design, Analysis, and Applications

Preface......Page 6
Other Polytopal Methods......Page 9
Foundations......Page 14
Applications to Advanced Models......Page 19
HHO Libraries......Page 23
Acknowledgements......Page 24
Contents......Page 25
Part I Foundations......Page 32
1.1 Mesh......Page 33
1.1.1 Polytopal Mesh......Page 34
1.1.2 Regular Mesh Sequence......Page 37
1.1.3 Geometric Bounds on Regular Mesh Sequences......Page 39
1.2 Function Spaces......Page 41
1.2.1.1 Lebesgue Spaces......Page 42
1.2.1.2 Sobolev Spaces......Page 43
1.2.2 Broken Sobolev Spaces......Page 45
1.2.3.1 The Polynomial Space Pnl......Page 49
1.2.3.2 Local and Broken Polynomial Spaces......Page 50
1.2.5 Lebesgue and Sobolev Embeddings in Local Polynomial Spaces......Page 51
1.2.6 Local Trace Inequalities on Regular Mesh Sequences......Page 56
1.3.1 Definition and Examples......Page 59
1.3.2 Approximation Properties of Bounded Projectors on Local Polynomial Spaces......Page 62
1.3.3 Approximation Properties of the Local L2-Orthogonal and Elliptic Projectors......Page 66
1.4.1 Approximation by Local Polynomials......Page 70
1.4.2 The Case of Mesh Elements and Faces......Page 72
2 Basic Principles of Hybrid High-Order Methods: The Poisson Problem......Page 75
2.1.1 Computing the Local Elliptic Projection from L2-Projections......Page 77
2.1.2 Local Space of Discrete Unknowns......Page 78
2.1.3 Potential Reconstruction Operator......Page 79
2.1.4 Local Contribution......Page 81
2.2.1 Global Space of Discrete Unknowns......Page 87
2.2.2 A Discrete Poincaré Inequality......Page 88
2.2.3 Global Bilinear Form......Page 90
2.2.4 Discrete Problem and Well-Posedness......Page 92
2.2.5 Flux Formulation......Page 93
2.2.5.2 Flux Formulation for an Abstract HHO Scheme......Page 94
2.2.5.3 Flux Formulation for the HHO Approximation of the Poisson Problem......Page 96
2.3.1 Energy Error Estimate......Page 98
2.3.2 Convergence of the Jumps......Page 101
2.3.3 L2-Error Estimate......Page 102
2.4 Other Boundary Conditions......Page 107
2.5.1 Two-Dimensional Test Case......Page 108
2.5.2 Three-Dimensional Test Case......Page 109
3 Variable Diffusion and Diffusion–Advection–Reaction......Page 112
3.1 Variable Diffusion......Page 113
3.1.1 Compliant Mesh Sequence......Page 114
3.1.2 The Oblique Elliptic Projector......Page 115
3.1.3.1 Diffusion-Dependent Local Potential Reconstruction......Page 119
3.1.3.2 Local Contribution......Page 121
3.1.4.1 Global Bilinear Form......Page 127
3.1.4.2 Discrete Problem......Page 130
3.1.4.3 Flux Formulation......Page 131
3.1.4.4 Energy Error Estimate......Page 132
3.2 Diffusion–Advection–Reaction......Page 134
3.2.1 Discretisation of Advective Terms with Upwind Stabilisation......Page 135
3.2.1.1 Reconstructed Advective Derivative......Page 136
3.2.1.2 Local Advective–Reactive Bilinear Form......Page 139
3.2.1.3 Global Advective–Reactive Bilinear Form......Page 140
3.2.2.2 Flux Formulation......Page 148
3.2.2.3 Initial Convergence Result......Page 151
3.2.3 Robust Convergence Including the Advective Derivative......Page 153
3.2.4 L2-Error Estimate......Page 160
3.3.1 Two-Dimensional Test Case......Page 169
3.3.2 Three-Dimensional Test Case......Page 173
4.1 A Posteriori Error Analysis......Page 176
4.1.1 Energy Error Upper Bound......Page 177
4.1.2 Energy Error Lower Bounds......Page 183
4.1.3 Numerical Examples: A Posteriori-Driven MeshAdaptivity......Page 189
4.1.3.1 Adaptively Refined Matching Tetrahedral Meshes......Page 190
4.1.3.2 Adaptive Mesh Coarsening......Page 191
4.2 Locally Variable Diffusion......Page 193
4.2.1 Discrete Gradient......Page 194
4.2.2 Local and Global Bilinear Forms......Page 196
4.2.3 Discrete Problem and Flux Formulation......Page 202
4.2.4 Energy Error Estimate......Page 204
4.2.5 L2-Error Estimate......Page 207
4.2.6 Numerical Tests......Page 211
5.1 Enrichment and Depletion of Element Unknowns......Page 214
5.1.1 Local Space and Interpolator......Page 215
5.1.2 Modified Elliptic Projector......Page 218
5.1.3 Potential Reconstruction......Page 222
5.1.4 Local Bilinear Form......Page 223
5.1.5 Discrete Problem and Energy Error Estimate......Page 225
5.1.6 Link with Hybridisable Discontinuous Galerkin  Methods......Page 228
5.1.7 L2-Error Analysis......Page 229
5.1.8.1 Two-Dimensional Test Case......Page 235
5.1.8.2 Three-Dimensional Test Case......Page 238
5.2.2 Properties of the Low-Order Potential Reconstruction on Simplices......Page 241
5.2.3 Link with HHO(0,-1)......Page 244
5.3.1 The HMM Method......Page 245
5.3.2 Equivalence Between HMM and HHO with k=0......Page 246
5.4.1 The Poisson Problem in Mixed Formulation......Page 249
5.4.2 Local Spaces of Discrete Unknowns......Page 250
5.4.3 Local Divergence and Flux Reconstructions......Page 251
5.4.4 Local Bilinear Forms......Page 252
5.4.5 Global Spaces of Discrete Unknownsand Discrete Problem......Page 253
5.4.6.1 Hybridisation......Page 254
5.4.6.2 Potential-to-Flux Operator......Page 255
5.4.6.3 Equivalence of the Mixed, Mixed Hybrid and Primal Formulations......Page 258
5.4.7 Link with the HHO Method......Page 259
5.5 Virtual Elements......Page 260
5.5.1 Local Virtual Space......Page 261
5.5.2 Virtual Reformulation of the Local HHO  Bilinear Form......Page 263
5.5.3 Global Virtual Space and Global Bilinear Form......Page 264
5.5.4 Virtual Reformulation of the HHO Scheme......Page 265
5.5.5 Link with Nonconforming Virtual Elements......Page 266
5.5.6 The Conforming Virtual Element Method......Page 267
5.5.6.1 Local Space and Interpolator......Page 268
5.5.6.2 Boundedness of the Interpolator and Approximation Properties of the Projector......Page 271
5.5.6.3 A Conforming Virtual Elements scheme for the Poisson Problem......Page 279
5.6.1 The Gradient Discretisation Method......Page 283
5.6.2 Discontinuous Skeletal Gradient Discretisations......Page 286
5.6.3.1 A Key Property and a Lifting of Face Differences......Page 290
5.6.3.2 A Stabilisation Term Based on a Local Raviart–Thomas–Nédélec Space......Page 293
5.6.4 Properties of Discontinuous Skeletal Gradient Discretisations......Page 296
Part II Applications to Advanced Models......Page 300
6 p-Laplacian and Leray–Lions......Page 301
6.1 Model......Page 302
6.2 Discrete Problem......Page 304
6.2.1 Discrete W1,p Space and Discrete Functional Analysis......Page 305
6.2.2.1 Gradient Reconstruction-Based W1,p-norm......Page 308
6.2.2.3 Equivalence of Reconstruction-Based W1,p-norms......Page 309
6.2.3 Discrete Problem and Well-Posedness......Page 312
6.2.4 Flux Formulation......Page 314
6.3 Error Estimates for the p-Laplacian......Page 316
6.3.1 Statement of the Error Estimates......Page 317
6.3.2 Consistency of the Stabilisation Function......Page 319
6.3.3 Strong Monotonicity and Continuity of the p-Laplace Flux Function......Page 322
6.3.4 Proof of the Error Estimates......Page 326
6.4 Convergence by Compactness for General Leray–Lions  Operators......Page 334
6.5.1 Mapping High-Order Unknowns to Lowest-Order Unknowns on Simplicial Submeshes......Page 341
6.5.2 Discrete Sobolev–Poincaré–Wirtinger Embeddings......Page 345
6.5.3 Discrete Trace Inequality......Page 346
6.5.4 Discrete Compactness......Page 348
6.6 Discrete Functional Analysis for Homogeneous Dirichlet Boundary Conditions......Page 351
7 Linear Elasticity......Page 353
7.1 Model......Page 354
7.1.2 Symmetric and Skew-Symmetric Gradients, Rigid-Body Motions......Page 355
7.1.3 The Elasticity Problem......Page 357
7.1.4 Weak Formulation and Well-Posedness......Page 358
7.2 Local Construction......Page 360
7.2.1 Regular Mesh Sequence with Star-Shaped Elements......Page 361
7.2.2 The Strain Projector......Page 362
7.2.3 Two Inspiring Relations......Page 364
7.2.4 Local Space of Discrete Unknowns......Page 366
7.2.5 Local Reconstructions......Page 368
7.2.6 Local Contribution......Page 370
7.3.1 Global Space of Discrete Unknowns......Page 375
7.3.2 Global Discrete Korn Inequalities in Broken Polynomial and HHO Spaces......Page 376
7.3.3 Global Bilinear Form......Page 380
7.3.4 Discrete Problem and Well-Posedness......Page 382
7.3.5 Flux Formulation......Page 383
7.4.1 Energy Error Estimate......Page 384
7.4.2 L2-Error Estimate......Page 386
7.4.3 Robustness in the Quasi-Incompressible Limit......Page 388
7.4.4 Numerical Examples......Page 390
7.5 Other Boundary Conditions......Page 391
7.6.1 A Global Discrete Strain Norm Including Jumps......Page 393
7.6.2 A Global Bilinear Form with Jump Penalisation......Page 394
7.6.3 Discrete Problem and Energy Error Estimate......Page 397
7.6.4.1 Quasi-Incompressible Test Case......Page 398
7.6.4.2 Singular Test Case......Page 399
7.7 Proof of the Uniform Local Second Korn Inequality......Page 401
8 Stokes......Page 408
8.1.1 The Stokes Problem......Page 409
8.1.2 Weak Formulation......Page 410
8.1.3 Inf–Sup Stability of the Pressure–Velocity Coupling......Page 411
8.2.1 Local Space of Discrete Velocity Unknowns......Page 414
8.2.2 Velocity and Divergence Reconstructions......Page 415
8.3.1 Global Spaces of Discrete Unknowns......Page 417
8.3.2 Viscous Term......Page 418
8.3.3 Pressure–Velocity Coupling......Page 420
8.3.4 Discrete Problem and Well-Posedness......Page 423
8.4 Flux Formulation......Page 425
8.5.1 Energy Error Estimate......Page 428
8.5.2 Improved L2-Error Estimates for the Velocity......Page 430
8.5.3 Other Hybrid Methods......Page 435
8.5.4 Numerical Example......Page 436
8.6.1 A Key Remark......Page 437
8.6.2 An Abstract Modification of the Right-Hand Side......Page 438
8.6.3 Pressure-Robust Error Estimate......Page 439
8.6.4 A Discretisation of Body Forces Based on a Raviart–Thomas–Nédélec Velocity Reconstruction......Page 440
8.6.4.2 An H(div;Ω)-Conforming Velocity Reconstruction......Page 441
8.6.4.3 Pressure-Robust Bilinear Form h......Page 442
8.6.5.1 Viscosity-Independence......Page 444
8.6.5.2 Convergence......Page 445
9 Navier–Stokes......Page 448
9.1.1 The Navier–Stokes Problem......Page 449
9.1.2 Weak Formulation......Page 450
9.1.3 Non-dissipativity of the Convective Term......Page 451
9.2.1 Discrete Problem and Design Properties for the Discrete Trilinear Form......Page 452
9.2.2 Existence and Uniqueness of a Discrete Solution......Page 454
9.3 Energy Error Estimate for Small Data......Page 458
9.4 Convective Stabilisation......Page 463
9.5.1 A Local Gradient Reconstruction......Page 464
9.5.2.1 A Gradient-Based Discrete Directional Derivative......Page 467
9.5.2.2 Discrete Trilinear Form......Page 470
9.5.3.1 Discrete Directional Derivative......Page 473
9.5.3.2 Discrete Divergence and Integration by Parts Formula......Page 476
9.5.3.3 Discrete Trilinear Form......Page 478
9.5.3.4 Flux Formulation......Page 482
9.6.1 Discrete Compactness and Strong Convergence of the Interpolates......Page 487
9.6.2 Convergence by Compactness......Page 490
9.7.1 Kovasznay Flow......Page 494
9.7.2 Lid-Driven Cavity Flow......Page 495
A.1.1 Setting......Page 502
A.1.2 Third Strang Lemma......Page 503
A.1.3 Aubin–Nitsche Trick......Page 505
A.2.1 Setting......Page 507
A.2.2 Stability and Energy Error Estimate......Page 508
A.2.3 Improved Error Estimate in a Weaker Norm......Page 510
B.1 Polynomial Bases and Degrees of Freedom......Page 512
B.1.1 Choice of Basis Functions......Page 513
B.2.1 Local Potential Reconstruction Operator......Page 515
B.2.2 Difference Operators......Page 518
B.2.3 Local Contribution......Page 519
B.3 Discrete Problem......Page 520
B.3.1 Assembly and Enforcement of Boundary Conditions......Page 521
B.3.2 Static Condensation......Page 522
References......Page 524
Model Index......Page 539
General Index......Page 541
Author Index......Page 547