Numerical Approximation of Partial Differential Equations

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ISBN: 9783540852674 | 3540852670
Cover: Paperback
Copyright: 9/1/2008

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This book deals with the numerical approximation of partial differential equations. Its scope is to provide a thorough illustration of numerical methods, carry out their stability and convergence analysis, derive error bounds, and discuss the algorithmic aspects relative to their implementation. A sound balancing of theoretical analysis, description of algorithms and discussion of applications is one of its main features. Many kinds of problems are addressed. A comprehensive theory of Galerkin method and its variants, as well as that of collocation methods, are developed for the spatial discretization. These theories are then specified to two numerical subspace realizations of remarkable interest: the finite element method and the spectral method. Book jacket.

Basic Concepts and Methods for PDEs' Approximation
Introduction	p. 1
The Conceptual Path Behind the Approximation	p. 2
Preliminary Notation and Function Spaces	p. 4
Some Results About Sobolev Spaces	p. 10
Comparison Results	p. 13
Numerical Solution of Linear Systems	p. 17
Direct Methods	p. 17
Banded Systems	p. 22
Error Analysis	p. 23
Generalities on Iterative Methods	p. 26
Classical Iterative Methods	p. 29
Jacobi Method	p. 29
Gauss-Seidel Method	p. 31
Relaxation Methods (S.O.R. and S.S.O.R.)	p. 32
Chebyshev Acceleration Method	p. 34
The Alternating Direction Iterative Method	p. 37
Modern Iterative Methods	p. 39
Preconditioned Richardson Method	p. 39
Conjugate Gradient Method	p. 46
Preconditioning	p. 51
Conjugate Gradient and Lanczos like Methods for Non-Symmetric Problems	p. 57
GCR, Orthomin and Orthodir Iterations	p. 57
Arnoldi and GMRES Iterations	p. 59
Bi-CG, CGS and Bi-CGSTAB Iterations	p. 62
The Multi-Grid Method	p. 65
The Multi-Grid Cycles	p. 65
A Simple Example	p. 67
Convergence	p. 70
Complements	p. 71
Finite Element Approximation	p. 73
Triangulation	p. 73
Piecewise-Polynomial Subspaces	p. 74
The Scalar Case	p. 75
The Vector Case	p. 76
Degrees of Freedom and Shape Functions	p. 77
The Scalar Case: Triangular Finite Elements	p. 77
The Scalar Case: Parallelepipedal Finite Elements	p. 80
The Vector Case	p. 82
The Interpolation Operator	p. 85
Interpolation Error: the Scalar Case	p. 85
Interpolation Error: the Vector Case	p. 91
Projection Operators	p. 96
Complements	p. 99
Polynomial Approximation	p. 101
Orthogonal Polynomials	p. 101
Gaussian Quadrature and Interpolation	p. 103
Chebyshev Expansion	p. 105
Chebyshev Polynomials	p. 105
Chebyshev Interpolation	p. 107
Chebyshev Projections	p. 113
Legendre Expansion	p. 115
Legendre Polynomials	p. 115
Legendre Interpolation	p. 117
Legendre Projections	p. 120
Two-Dimensional Extensions	p. 121
The Chebyshev Case	p. 121
The Legendre Case	p. 124
Complements	p. 127
Galerkin, Collocation and Other Methods	p. 129
An Abstract Reference Boundary Value Problem	p. 129
Some Results of Functional Analysis	p. 133
Galerkin Method	p. 136
Petrov-Galerkin Method	p. 138
Collocation Method	p. 140
Generalized Galerkin Method	p. 141
Time-Advancing Methods for Time-Dependent Problems	p. 144
Semi-Discrete Approximation	p. 148
Fully-Discrete Approximation	p. 148
Fractional-Step and Operator-Splitting Methods	p. 151
Complements	p. 156
Approximation of Boundary Value Problems
Elliptic Problems: Approximation by Galerkin and Collocation Methods	p. 159
Problem Formulation and Mathematical Properties	p. 159
Variational Form of Boundary Value Problems	p. 161
Existence, Uniqueness and A-Priori Estimates	p. 164
Regularity of Solutions	p. 167
On the Degeneracy of the Constants in Stability and Error Estimates	p. 168
Numerical Methods: Construction and Analysis	p. 169
Galerkin Method: Finite Element and Spectral Approximations	p. 170
Spectral Collocation Method	p. 179
Generalized Galerkin Method	p. 187
Algorithmic Aspects	p. 189
Algebraic Formulation	p. 190
The Finite Element Case	p. 192
The Spectral Collocation Case	p. 198
Domain Decomposition Methods	p. 204
The Schwarz Method	p. 206
Iteration-by-Subdomain Methods Based on Transmission Conditions at the Interface	p. 209
The Steklov-Poincare Operator	p. 212
The Connection Between Iterations-by-Subdomain Methods and the Schur Complement System	p. 215
Elliptic Problems: Approximation by Mixed and Hybrid Methods	p. 217
Alternative Mathematical Formulations	p. 217
The Minimum Complementary Energy Principle	p. 218
Saddle-Point Formulations: Mixed and Hybrid Methods	p. 222
Approximation by Mixed Methods	p. 230
Setting up and Analysis	p. 230
An Example: the Raviart-Thomas Finite Elements	p. 235
Some Remarks on the Algorithmic Aspects	p. 241
The Approximation of More General Constrained Problems	p. 246
Abstract Formulation	p. 246
Analysis of Stability and Convergence	p. 250
How to Verify the Uniform Compatibility Condition	p. 253
Complements	p. 255
Steady Advection-Diffusion Problems	p. 257
Mathematical Formulation	p. 257
A One-Dimensional Example	p. 258
Galerkin Approximation and Centered Finite Differences	p. 259
Upwind Finite Differences and Numerical Diffusion	p. 262
Spectral Approximation	p. 263
Stabilization Methods	p. 265
The Artificial Diffusion Method	p. 267
Strongly Consistent Stabilization Methods for Finite Elements	p. 269
Stabilization by Bubble Functions	p. 273
Stabilization Methods for Spectral Approximation	p. 277
Analysis of Strongly Consistent Stabilization Methods	p. 280
Some Numerical Results	p. 288
The Heterogeneous Method	p. 289
The Stokes Problem	p. 297
Mathematical Formulation and Analysis	p. 297
Galerkin Approximation	p. 300
Algebraic Form of the Stokes Problem	p. 303
Compatibility Condition and Spurious Pressure Modes	p. 304
Divergence-Free Property and Locking Phenomena	p. 305
Finite Element Approximation	p. 306
Discontinuous Pressure Finite Elements	p. 306
Continuous Pressure Finite Elements	p. 310
Stabilization Procedures	p. 311
Approximation by Spectral Methods	p. 317
Spectral Galerkin Approximation	p. 319
Spectral Collocation Approximation	p. 323
Spectral Generalized Galerkin Approximation	p. 324
Solving the Stokes System	p. 325
The Pressure-Matrix Method	p. 326
The Uzawa Method	p. 327
The Arrow-Hurwicz Method	p. 328
Penalty Methods	p. 329
The Augmented-Lagrangian Method	p. 330
Methods Based on Pressure Solvers	p. 331
A Global Preconditioning Technique	p. 335
Complements	p. 337
The Steady Navier-Stokes Problem	p. 339
Mathematical Formulation	p. 339
Other Kind of Boundary Conditions	p. 343
An Abstract Formulation	p. 345
Finite Dimensional Approximation	p. 346
An Abstract Approximate Problem	p. 347
Approximation by Mixed Finite Element Methods	p. 349
Approximation by Spectral Collocation Methods	p. 351
Numerical Algorithms	p. 353
Newton Methods and the Continuation Method	p. 353
An Operator-Splitting Algorithm	p. 358
Stream Function-Vorticity Formulation of the Navier-Stokes Equations	p. 359
Complements	p. 361
Approximation of Initial-Boundary Value Problems
Parabolic Problems	p. 363
Initial-Boundary Value Problems and Weak Formulation	p. 363
Mathematical Analysis of Initial-Boundary Value Problems	p. 365
Semi-Discrete Approximation	p. 373
The Finite Element Case	p. 373
The Case of Spectral Methods	p. 379
Time-Advancing by Finite Differences	p. 384
The Finite Element Case	p. 385
The Case of Spectral Methods	p. 396
Some Remarks on the Algorithmic Aspects	p. 401
Complements	p. 404
Unsteady Advection-Diffusion Problems	p. 405
Mathematical Formulation	p. 405
Time-Advancing by Finite Differences	p. 408
A Sharp Stability Result for the [theta]-scheme	p. 408
A Semi-Implicit Scheme	p. 411
The Discontinuous Galerkin Method for Stabilized Problems	p. 415
Operator-Splitting Methods	p. 418
A Characteristic Galerkin Method	p. 423
The Unsteady Navier-Stokes Problem	p. 429
The Navier-Stokes Equations for Compressible and Incompressible Flows	p. 430
Compressible Flows	p. 431
Incompressible Flows	p. 432
Mathematical Formulation and Behaviour of Solutions	p. 433
Semi-Discrete Approximation	p. 434
Time-Advancing by Finite Differences	p. 438
Operator-Splitting Methods	p. 441
Other Approaches	p. 446
Complements	p. 448
Hyperbolic Problems	p. 449
Some Instances of Hyperbolic Equations	p. 450
Linear Scalar Advection Equations	p. 450
Linear Hyperbolic Systems	p. 451
Initial-Boundary Value Problems	p. 453
Nonlinear Scalar Equations	p. 455
Approximation by Finite Differences	p. 461
Linear Scalar Advection Equations and Hyperbolic Systems	p. 461
Stability, Consistency, Convergence	p. 465
Nonlinear Scalar Equations	p. 471
High Order Shock Capturing Schemes	p. 475
Approximation by Finite Elements	p. 481
Galerkin Method	p. 482
Stabilization of the Galerkin Method	p. 485
Space-Discontinuous Galerkin Method	p. 487
Schemes for Time-Discretization	p. 488
Approximation by Spectral Methods	p. 490
Spectral Collocation Method: the Scalar Case	p. 491
Spectral Collocation Method: the Vector Case	p. 494
Time-Advancing and Smoothing Procedures	p. 496
Second Order Linear Hyperbolic Problems	p. 497
The Finite Volume Method	p. 501
Complements	p. 508
References	p. 509
Subject Index	p. 537
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