Analysis and Numerics of Partial Differential Equations

Cover......Page 1
Analysis and Numerics of Partial Differential Equations......Page 3
Preface......Page 6
Contents......Page 7
List of Contributors......Page 9
Personal Memories......Page 12
1 The Beginnings in Modena......Page 14
2 The Years in Genoa with G. Stampacchia......Page 16
3 The Collaboration with J.-L. Lions in the Study of Boundary Value Problems......Page 17
References......Page 19
Enrico Magenes and the Dam Problem......Page 21
1 Introduction......Page 27
2 Cardiac Body Surface Maps and Inverse Potential Problems......Page 28
3 Cardiac Excitation Sources Models and Inverse Excitation Wavefront Problems......Page 33
References......Page 44
1 Stefan Problems and Semigroups: Analysis and Numerical Approximation......Page 45
2 Stefan Problems in a Concentrated Capacity......Page 48
3 Approximation of Interfaces, Adaptivity and Applications......Page 49
References......Page 50
1 Introduction......Page 54
2 Magenes and the Teaching of Mathematics: A Constant Commitment......Page 55
3 Magenes and the Research in Mathematics Education......Page 56
4 Magenes and the Research Unit in Pavia: The Project \"Mathematics as a Discovery\"......Page 57
5 Two Talks as Keynote Speaker......Page 58
References......Page 59
References......Page 60
1 Introduction......Page 67
2.1 Absolutely Continuous Curves and Slopes......Page 69
2.2 The Space (P(X),W2)......Page 70
2.3 Geodesically Convex Functionals and Their Gradient Flows......Page 71
Energy Dissipation Equality......Page 73
Evolution Variational Inequality......Page 74
3 Hopf-Lax Formula and Hamilton-Jacobi Equation......Page 75
4 Weak Definitions of Gradient......Page 79
4.1 The \"Vertical\" Approach: Minimal Relaxed Slope......Page 80
Laplacian: Definition and Basic Properties......Page 82
4.2 The \"Horizontal\" Approach: Weak Upper Gradients......Page 85
Negligible Sets of Curves and Functions Sobolev Along a.e. Curve......Page 86
A Bound from Below on Weak Gradients......Page 90
4.3 The Two Notions of Gradient Coincide......Page 92
4.4 Comparison with Previous Approaches......Page 95
5 The Relative Entropy and Its W2-Gradient Flow......Page 98
6 The Heat Flow as Gradient Flow......Page 103
7 A Metric Brenier Theorem......Page 106
8.1 On Horizontal and Vertical Derivatives Again......Page 110
8.2 Two Important Formulas......Page 112
9 Riemannian Ricci Bounds......Page 114
References......Page 118
1 Introduction......Page 120
2.1 The Assembly of Finite Element Spaces......Page 122
2.2 The Lagrange Finite Element Family......Page 123
2.3 Exterior Calculus......Page 125
3 Families of Finite Element Differential Forms on Simplicial Meshes......Page 127
3.2 The Koszul Complex......Page 128
3.3 The Polynomial Space Pr-Lambdak......Page 130
3.4 The Pr-Lambdak(Th) Family of Finite Element Differential Forms......Page 131
3.6 Historical Notes......Page 133
4.1 The Qr-Lambdak Family......Page 136
4.2 A Second Family of Finite Element Differential Forms on Cubes......Page 138
References......Page 142
1 Introduction......Page 144
2 Fourier Multipliers of K......Page 147
3 Fourier Analysis......Page 150
4 Kernel Estimates......Page 155
5 Gradient Bounds......Page 158
6 Proof of Theorem 1.2......Page 159
References......Page 166
1 Introduction......Page 167
2 From Approximation Theory to a-Priori Adaptive hp Methods......Page 169
2.1 An Elementary Analysis of hp Approximations over Dyadic Partitions......Page 172
The Case of an Algebraic Singularity......Page 173
The Case of a Piecewise-Analytic Function......Page 174
3 hp Adaptivity......Page 175
3.1 A-Posteriori hp Error Estimates......Page 176
3.2 Adaptive hp Methods......Page 177
3.3 Convergence of Adaptive Spectral/hp Methods......Page 179
4 Spectral Adaptive Algorithms with Optimality Properties......Page 180
4.1 Bases and Norm Representations......Page 181
Algebraic Representation and Properties of the Stiffness Matrix......Page 183
4.2 The Constitutive Elements of an Adaptive Algorithm......Page 184
4.3 An Adaptive Algorithm with Convergence Rate......Page 186
4.4 Nonlinear Approximation in Gevrey Spaces......Page 187
4.5 Complexity Analysis of the Algorithm......Page 190
References......Page 192
A Theory and Challenges for Coarsening in Microstructure......Page 195
1 Introduction......Page 196
2 Reprise of Mesoscale Theory......Page 198
3 Discussion of the Simulation......Page 200
4 A Simplified Coarsening Model with Entropy and Dissipation......Page 202
4.1 Formulation......Page 204
4.2 The Mass Transport Paradigm......Page 207
5 Validation of the Scheme......Page 210
6.1 Quadratic Interfacial Energy Density......Page 212
6.2 Quartic Interfacial Energy Density......Page 214
6.3 Remarks on a Theory for the Diffusion Coefficient sigma or the Temperature-Like Parameter......Page 216
7 Closing Comments......Page 217
References......Page 219
1 Introduction......Page 223
2 Generalized Empirical Interpolation Method......Page 225
2.1 Recall of the Empirical Interpolation Method......Page 226
2.2 The Generalization......Page 227
2.3 Numerical Results......Page 230
3.1 The Framework......Page 233
3.2 The Combined Approach-Numerical Results......Page 234
4 About Noisy Data......Page 235
References......Page 236
1 Introduction......Page 238
2 Elliptic Operators with Fractal Singularities......Page 241
3 Elliptic Operators with Fractal Degeneracy......Page 249
4 Interfacial Heat Transmission......Page 251
References......Page 255
1 Introduction......Page 257
2.1 Biomembranes: Modeling and Simulations......Page 259
Fluid Membranes......Page 260
2.2 The Laplace-Beltrami Operator and Curvature......Page 262
2.3 Surface Diffusion and Epitaxial Films......Page 264
2.4 Geometrically Consistent Accuracy Preserving Algorithm......Page 266
3.1 Representation of Parametric Surfaces......Page 267
3.2 Interpolation of Parametric Surfaces......Page 269
4.1 Basic Differential Geometry......Page 271
4.2 Variational Formulation and Galerkin Method......Page 273
5.1 Geometric Error and Estimator......Page 276
5.2 Upper and Lower Bounds for the Energy Error......Page 279
5.3 Properties of the PDE Estimator and Data Oscillation......Page 283
6.1 Module ADAPT_SURFACE......Page 286
Procedure ESTIMATE......Page 287
Procedure REFINE......Page 288
7 Conditional Contraction Property......Page 289
8.1 Approximation Classes......Page 292
8.2 Convergence Rates......Page 293
8.3 Greedy Algorithm......Page 298
9 Asymptotics: Role of omega......Page 299
9.2 Case alpha=2/5......Page 301
10 Conclusions and Comments......Page 303
References......Page 304
Generalized Reduced Basis Methods and n-Width Estimates for the Approximation of the Solution Manifold of Parametric PDEs......Page 307
1 Introduction......Page 308
2 Generalized Reduced Basis Method for Uniform Approximation of µ-PDEs......Page 311
3 Approximation Theoretical Basis for the Generalized Reduced Basis Method......Page 317
4 An Extended Result of Exponential Convergence......Page 319
5 Numerical Example of a Parameter-Dependent Diffusion Problem......Page 324
6 Conclusions......Page 327
References......Page 328
Foreword......Page 330
Stefan-Type Problems......Page 331
A Doubly Nonlinear Equation......Page 332
Plan of Work......Page 333
Literature......Page 334
Phase Relaxation......Page 335
Glass Formation......Page 336
Eckart\'s Theory of Nonequilibrium Thermodynamics......Page 338
Balance Laws and Gibbs-Type Formula......Page 339
Phenomenological Laws and Phase Relaxation......Page 341
Potential Structure of the Phenomenological Laws......Page 342
4 Weak Formulation and Existence Theorems......Page 343
The Fitzpatrick Theorem......Page 348
Gamma-Compactness and Stability of Representative Functions......Page 349
Compactness and Structural Stability......Page 350
Maximal Monotone Flows......Page 351
Variational Formulations......Page 352
Compactness of Representative Functions......Page 353
Tartar\'s Example......Page 354
Asymptotic Short Memory......Page 355
Variational Formulations of a Doubly Nonlinear Flow......Page 356
Conclusions as for the Variational Formulation of (128)......Page 358
Variational Formulation of the Single-Phase Problem......Page 359
References......Page 361