Finite Elements III

417 Pages·2021·10.686 MB·English

by  Alexandre Ern, Jean-Luc Guermond

Texts in Applied Mathematics 74
Alexandre Ern
Jean-Luc Guermond
Finite Elements III
First-Order and Time-Dependent PDEs
Texts in Applied Mathematics
Volume 74
Editors-in-Chief
Anthony Bloch, University of Michigan, Ann Arbor, MI, USA
Charles L. Epstein, University of Pennsylvania, Philadelphia, PA, USA
Alain Goriely, University of Oxford, Oxford, UK
Leslie Greengard, New York University, New York, NY, USA
Series Editors
J. Bell, Lawrence Berkeley National Laboratory, Berkeley, CA, USA
R. Kohn, New York University, New York, NY, USA
P. Newton, University of Southern California, Los Angeles, CA, USA
C. Peskin, New York University, New York, NY, USA
R. Pego, Carnegie Mellon University, Pittsburgh, PA, USA
L. Ryzhik, Stanford University, Stanford, CA, USA
A. Singer, Princeton University, Princeton, NJ, USA
A Stevens, University of Münster, Münster, Germany
A. Stuart, University of Warwick, Coventry, UK
T. Witelski, Duke University, Durham, NC, USA
S. Wright, University of Wisconsin, Madison, WI, USA
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Alexandre Ern Jean-Luc Guermond
(cid:129)
Finite Elements III
First-Order and Time-Dependent PDEs
123
Alexandre Ern Jean-Luc Guermond
CERMICS Department ofMathematics
Ecoledes Ponts andINRIA Paris Texas A&MUniversity
Marnela ValléeandParis, France CollegeStation, TX,USA
ISSN 0939-2475 ISSN 2196-9949 (electronic)
Textsin AppliedMathematics
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Contents
Part XII. First-order PDEs
56 Friedrichs’ systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
56.1 Basic ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
56.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
56.3 Weak formulation and well-posedness . . . . . . . . . . . . . . . . . 10
57 Residual-based stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
57.1 Model problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
57.2 Least-squares (LS) approximation . . . . . . . . . . . . . . . . . . . . 20
57.3 Galerkin/least-squares (GaLS). . . . . . . . . . . . . . . . . . . . . . . 22
57.4 Boundary penalty for Friedrichs’ systems . . . . . . . . . . . . . . 29
58 Fluctuation-based stabilization (I) . . . . . . . . . . . . . . . . . . . . . . . . . 35
58.1 Discrete setting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
58.2 Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
58.3 Continuous interior penalty . . . . . . . . . . . . . . . . . . . . . . . . . 39
58.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
59 Fluctuation-based stabilization (II) . . . . . . . . . . . . . . . . . . . . . . . . 47
59.1 Two-scale decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
59.2 Local projection stabilization . . . . . . . . . . . . . . . . . . . . . . . . 50
59.3 Subgrid viscosity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
59.4 Error analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
59.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
v
vi Contents
60 Discontinuous Galerkin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
60.1 Discrete setting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
60.2 Centered fluxes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
60.3 Tightened stability by jump penalty . . . . . . . . . . . . . . . . . . 63
61 Advection-diffusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
61.1 Model problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
61.2 Discrete setting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
61.3 Stability and error analysis . . . . . . . . . . . . . . . . . . . . . . . . . 74
61.4 Divergence-free advection. . . . . . . . . . . . . . . . . . . . . . . . . . . 80
62 Stokes equations: Residual-based stabilization. . . . . . . . . . . . . . . . 85
62.1 Model problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
62.2 Discrete setting for GaLS stabilization. . . . . . . . . . . . . . . . . 86
62.3 Stability and well-posedness. . . . . . . . . . . . . . . . . . . . . . . . . 88
62.4 Error analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
63 Stokes equations: Other stabilizations . . . . . . . . . . . . . . . . . . . . . . 95
63.1 Continuous interior penalty . . . . . . . . . . . . . . . . . . . . . . . . . 95
63.2 Discontinuous Galerkin . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
Part XIII. Parabolic PDEs
64 Bochner integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
64.1 Bochner integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
64.2 Weak time derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
65 Weak formulation and well-posedness . . . . . . . . . . . . . . . . . . . . . . 123
65.1 Weak formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
65.2 Well-posedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
65.3 Maximum principle for the heat equation . . . . . . . . . . . . . . 132
66 Semi-discretization in space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
66.1 Model problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
66.2 Principle and algebraic realization . . . . . . . . . . . . . . . . . . . . 136
66.3 Error analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
67 Implicit and explicit Euler schemes . . . . . . . . . . . . . . . . . . . . . . . . 147
67.1 Implicit Euler scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
67.2 Explicit Euler scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
68 BDF2 and Crank–Nicolson schemes. . . . . . . . . . . . . . . . . . . . . . . . 161
68.1 Discrete setting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
68.2 BDF2 scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
68.3 Crank–Nicolson scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
Contents vii
69 Discontinuous Galerkin in time . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
69.1 Setting for the time discretization . . . . . . . . . . . . . . . . . . . . 177
69.2 Formulation of the method. . . . . . . . . . . . . . . . . . . . . . . . . . 179
69.3 Stability and error analysis . . . . . . . . . . . . . . . . . . . . . . . . . 185
69.4 Algebraic realization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
70 Continuous Petrov–Galerkin in time . . . . . . . . . . . . . . . . . . . . . . . 195
70.1 Formulation of the method. . . . . . . . . . . . . . . . . . . . . . . . . . 195
70.2 Stability and error analysis . . . . . . . . . . . . . . . . . . . . . . . . . 201
70.3 Algebraic realization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
71 Analysis using inf-sup stability. . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
71.1 Well-posedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
71.2 Semi-discretization in space . . . . . . . . . . . . . . . . . . . . . . . . . 217
71.3 dG(k) scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
71.4 cPG(k) scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
Part XIV. Time-dependent Stokes equations
72 Weak formulations and well-posedness . . . . . . . . . . . . . . . . . . . . . 231
72.1 Model problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
72.2 Constrained weak formulation . . . . . . . . . . . . . . . . . . . . . . . 233
72.3 Mixed weak formulation with smooth data . . . . . . . . . . . . . 234
72.4 Mixed weak formulation with rough data . . . . . . . . . . . . . . 237
73 Monolithic time discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
73.1 Model problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
73.2 Space semi-discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
73.3 Implicit Euler approximation . . . . . . . . . . . . . . . . . . . . . . . . 248
73.4 Higher-order time approximation. . . . . . . . . . . . . . . . . . . . . 252
74 Projection methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
74.1 Model problem and Helmholtz decomposition . . . . . . . . . . . 255
74.2 Pressure correction in standard form . . . . . . . . . . . . . . . . . . 256
74.3 Pressure correction in rotational form . . . . . . . . . . . . . . . . . 261
74.4 Finite element approximation . . . . . . . . . . . . . . . . . . . . . . . 263
75 Artificial compressibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
75.1 Stability under compressibility perturbation . . . . . . . . . . . . 267
75.2 First-order artificial compressibility . . . . . . . . . . . . . . . . . . . 268
75.3 Higher-order artificial compressibility . . . . . . . . . . . . . . . . . 272
75.4 Finite element implementation. . . . . . . . . . . . . . . . . . . . . . . 275
viii Contents
Part XV. Time-dependent first-order linear PDEs
76 Well-posedness and space semi-discretization . . . . . . . . . . . . . . . . 279
76.1 Maximal monotone operators. . . . . . . . . . . . . . . . . . . . . . . . 279
76.2 Well-posedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
76.3 Time-dependent Friedrichs’ systems . . . . . . . . . . . . . . . . . . 286
76.4 Space semi-discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
77 Implicit time discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
77.1 Model problem and space discretization. . . . . . . . . . . . . . . . 295
77.2 Implicit Euler scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
77.3 Error analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
78 Explicit time discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
78.1 Explicit Runge–Kutta (ERK) schemes. . . . . . . . . . . . . . . . . 305
78.2 Explicit Euler scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310
78.3 Second-order two-stage ERK schemes . . . . . . . . . . . . . . . . . 312
78.4 Third-order three-stage ERK schemes . . . . . . . . . . . . . . . . . 318
Part XVI. Nonlinear hyperbolic PDEs
79 Scalar conservation equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
79.1 Weak and entropy solutions. . . . . . . . . . . . . . . . . . . . . . . . . 323
79.2 Riemann problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330
80 Hyperbolic systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
80.1 Weak solutions and examples. . . . . . . . . . . . . . . . . . . . . . . . 337
80.2 Riemann problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344
81 First-order approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
81.1 Scalar conservation equations. . . . . . . . . . . . . . . . . . . . . . . . 353
81.2 Hyperbolic systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
82 Higher-order approximation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367
82.1 Higher order in time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367
82.2 Higher order in space for scalar equations . . . . . . . . . . . . . . 374
83 Higher-order approximation and limiting . . . . . . . . . . . . . . . . . . . 383
83.1 Higher-order techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383
83.2 Limiting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388
References.... .... .... .... ..... .... .... .... .... .... ..... .... 401
Index .... .... .... .... .... ..... .... .... .... .... .... ..... .... 415
Part XII
First-order PDEs