Partial differential equations and the finite element method /
Pavel Solin.
Hoboken, N.J. : Wiley-Interscience, c2006.
xxvii, 472 p.
0471720704 (cloth : acid-free paper)
More Details
Hoboken, N.J. : Wiley-Interscience, c2006.
0471720704 (cloth : acid-free paper)
catalogue key
Includes bibliographical references and index.
A Look Inside
About the Author
Author Affiliation
Pavel Solin, PhD, is Associate Professor in the Department of Mathematical Sciences at The University of Texas at El Paso
Full Text Reviews
Appeared in Choice on 2006-07-01:
This well-written book discusses the modern methods of partial differential equations and the finite element method, containing detailed discussions of many of the algorithms of the latter. The basics of partial differential equations are included in chapter 1; chapters 2 through 4 are devoted to continuous and nodal elements. Solin (Univ. of Texas at El Paso) then covers higher-order methods for the numerical solution of ordinary differential equations in chapter 5. Chapter 6 explains elastic beam and plate bending problems through their fourth-order partial differential equations. The final chapter discusses the equations of electromagnetics. Background material in the first appendix includes an explanation of linear, normed, inner-product, and Sobolev spaces. A concluding appendix discusses sparse matrix solvers and Hermes software with examples. There are 185 figures, 20 tables, numerous examples, and 185 well-chosen exercises. Summing Up: Recommended. Upper-division undergraduates through professionals. D. P. Turner Faulkner University
Review Quotes
"This well-written book discusses the modern methods of partial differential equations and the finite element methods...recommended." ( CHOICE , July 2006)
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Choice, July 2006
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Bowker Data Service Summary
'Partial Differential Equations and the Finite Element Method' provides a much-needed, clear, and systematic introduction to modern theory of partial differential equations (PDEs) and finite element methods (FEM). Both nodal and hierachic concepts of the FEM are examined.
Back Cover Copy
A systematic introduction to partial differential equations and modern finite element methods for their efficient numerical solution Partial Differential Equations and the Finite Element Method provides a much-needed, clear, and systematic introduction to modern theory of partial differential equations (PDEs) and finite element methods (FEM). Both nodal and hierachic concepts of the FEM are examined. Reflecting the growing complexity and multiscale nature of current engineering and scientific problems, the author emphasizes higher-order finite element methods such as the spectral or hp-FEM. A solid introduction to the theory of PDEs and FEM contained in Chapters 1-4 serves as the core and foundation of the publication. Chapter 5 is devoted to modern higher-order methods for the numerical solution of ordinary differential equations (ODEs) that arise in the semidiscretization of time-dependent PDEs by the Method of Lines (MOL). Chapter 6 discusses fourth-order PDEs rooted in the bending of elastic beams and plates and approximates their solution by means of higher-order Hermite and Argyris elements. Finally, Chapter 7 introduces the reader to various PDEs governing computational electromagnetics and describes their finite element approximation, including modern higher-order edge elements for Maxwell's equations. The understanding of many theoretical and practical aspects of both PDEs and FEM requires a solid knowledge of linear algebra and elementary functional analysis, such as functions and linear operators in the Lebesgue, Hilbert, and Sobolev spaces. These topics are discussed with the help of many illustrative examples in Appendix A, which is provided as a service for those readers who need to gain the necessary background or require a refresher tutorial. Appendix B presents several finite element computations rooted in practical engineering problems and demonstrates the benefits of using higher-order FEM. Numerous finite element algorithms are written out in detail alongside implementation discussions. Exercises, including many that involve programming the FEM, are designed to assist the reader in solving typical problems in engineering and science. Specifically designed as a coursebook, this student-tested publication is geared to upper-level undergraduates and graduate students in all disciplines of computational engineeringand science. It is also a practical problem-solving reference for researchers, engineers, and physicists.
Table of Contents
List of Figuresp. xv
List of Tablesp. xxi
Prefacep. xxiii
Acknowledgmentsp. xxv
Partial Differential Equationsp. 1
Selected general propertiesp. 2
Classification and examplesp. 2
Hadamard's well-posednessp. 5
General existence and uniqueness resultsp. 9
Exercisesp. 11
Second-order elliptic problemsp. 13
Weak formulation of a model problemp. 13
Bilinear forms, energy norm, and energetic inner productp. 16
The Lax-Milgram lemmap. 18
Unique solvability of the model problemp. 18
Nonhomogeneous Dirichlet boundary conditionsp. 19
Neumann boundary conditionsp. 21
Newton (Robin) boundary conditionsp. 22
Combining essential and natural boundary conditionsp. 23
Energy of elliptic problemsp. 24
Maximum principles and well-posednessp. 26
Exercisesp. 29
Second-order parabolic problemsp. 30
Initial and boundary conditionsp. 30
Weak formulationp. 30
Existence and uniqueness of solutionp. 31
Exercisesp. 32
Second-order hyperbolic problemsp. 33
Initial and boundary conditionsp. 33
Weak formulation and unique solvabilityp. 34
The wave equationp. 34
Exercisesp. 35
First-order hyperbolic problemsp. 36
Conservation lawsp. 36
Characteristicsp. 38
Exact solution to linear first-order systemsp. 39
Riemann problemp. 41
Nonlinear flux and shock formationp. 43
Exercisesp. 44
Continuous Elements for 1D Problemsp. 45
The general frameworkp. 45
The Galerkin methodp. 46
Orthogonality of error and Cea's lemmap. 49
Convergence of the Galerkin methodp. 50
Ritz method for symmetric problemsp. 51
Exercisesp. 51
Lowest-order elementsp. 51
Model problemp. 52
Finite-dimensional subspace V[subscript n subset or is implied by] Vp. 52
Piecewise-affine basis functionsp. 53
The system of linear algebraic equationsp. 54
Element-by-element assembling procedurep. 55
Refinement and convergencep. 56
Exercisesp. 57
Higher-order numerical quadraturep. 59
Gaussian quadrature rulesp. 59
Selected quadrature constantsp. 61
Adaptive quadraturep. 63
Exercisesp. 65
Higher-order elementsp. 66
Motivation problemp. 66
Affine concept: reference domain and reference mapsp. 67
Transformation of weak forms to the reference domainp. 69
Higher-order Lagrange nodal shape functionsp. 70
Chebyshev and Gauss-Lobatto nodal pointsp. 71
Higher-order Lobatto hierarchic shape functionsp. 74
Constructing basis of the space V[subscript h,p]p. 76
Data structuresp. 77
Assembling algorithmp. 79
Exercisesp. 82
The sparse stiffness matrixp. 84
Compressed sparse row (CSR) data formatp. 84
Condition numberp. 84
Conditioning of shape functionsp. 86
Stiffness matrix for the Lobatto shape functionsp. 88
Exercisesp. 89
Implementing nonhomogeneous boundary conditionsp. 89
Dirichlet boundary conditionsp. 89
Combination of essential and natural conditionsp. 91
Exercisesp. 92
Interpolation on finite elementsp. 93
The Hilbert space settingp. 93
Best interpolantp. 94
Projection-based interpolantp. 96
Nodal interpolantp. 99
Exercisesp. 102
General Concept of Nodal Elementsp. 103
The nodal finite elementp. 103
Unisolvency and nodal basisp. 104
Checking unisolvencyp. 106
Example: lowest-order Q[superscript 1]-and P[superscript 1]-elementsp. 107
Q[superscript 1]-elementp. 108
P[superscript 1]-elementp. 110
Invertibility of the quadrilateral reference map x[subscript K]p. 113
Interpolation on nodal elementsp. 114
Local nodal interpolantp. 115
Global interpolant and conformityp. 116
Conformity to the Sobolev space H[superscript 1]p. 119
Equivalence of nodal elementsp. 120
Exercisesp. 122
Continuous Elements for 2D Problemsp. 125
Lowest-order elementsp. 126
Model problem and its weak formulationp. 126
Approximations and variational crimesp. 127
Basis of the space V[subscript h,p]p. 129
Transformation of weak forms to the reference domainp. 131
Simplified evaluation of stiffness integralsp. 133
Connectivity arraysp. 134
Assembling algorithm for Q[superscript 1]/P[superscript 1]-elementsp. 135
Lagrange interpolation on Q[superscript 1]/P[superscript 1]-meshesp. 137
Exercisesp. 137
Higher-order numerical quadrature in 2Dp. 139
Gaussian quadrature on quadsp. 139
Gaussian quadrature on trianglesp. 139
Higher-order nodal elementsp. 142
Product Gauss-Lobatto pointsp. 142
Lagrange-Gauss-Lobatto Q[superscript p,r]-elementsp. 143
Lagrange interpolation and the Lebesgue constantp. 148
The Fekete pointsp. 149
Lagrange-Fekete P[superscript p]-elementsp. 152
Basis of the space V[subscript h,p]p. 154
Data structuresp. 157
Connectivity arraysp. 160
Assembling algorithm for Q[superscript p]/P[superscript p]-elementsp. 162
Lagrange interpolation on Q[superscript p]/P[superscript p]-meshesp. 166
Exercisesp. 166
Transient Problems and ODE Solversp. 167
Method of linesp. 168
Model problemp. 168
Weak formulationp. 168
The ODE systemp. 169
Construction of the initial vectorp. 170
Autonomous systems and phase flowp. 171
Selected time integration schemesp. 172
One-step methods, consistency and convergencep. 173
Explicit and implicit Euler methodsp. 175
Stiffnessp. 177
Explicit higher-order RK schemesp. 179
Embedded RK methods and adaptivityp. 182
General (implicit) RK schemesp. 184
Introduction to stabilityp. 185
Autonomization of RK methodsp. 186
Stability of linear autonomous systemsp. 187
Stability functions and stability domainsp. 188
Stability functions for general RK methodsp. 191
Maximum consistency order of IRK methodsp. 193
A-stability and L-stabilityp. 194
Higher-order IRK methodsp. 197
Collocation methodsp. 197
Gauss and Radau IRK methodsp. 200
Solution of nonlinear systemsp. 202
Exercisesp. 205
Beam and Plate Bending Problemsp. 209
Bending of elastic beamsp. 210
Euler-Bernoulli modelp. 210
Boundary conditionsp. 212
Weak formulationp. 214
Existence and uniqueness of solutionp. 214
Lowest-order Hermite elements in 1Dp. 216
Model problemp. 216
Cubic Hermite elementsp. 218
Higher-order Hermite elements in 1Dp. 220
Nodal higher-order elementsp. 220
Hierarchic higher-order elementsp. 222
Conditioning of shape functionsp. 225
Basis of the space V[subscript h,p]p. 226
Transformation of weak forms to the reference domainp. 228
Connectivity arraysp. 228
Assembling algorithmp. 231
Interpolation on Hermite elementsp. 233
Hermite elements in 2Dp. 236
Lowest-order elementsp. 236
Higher-order Hermite-Fekete elementsp. 238
Design of basis functionsp. 240
Global nodal interpolant and conformityp. 242
Bending of elastic platesp. 242
Reissner-Mindlin (thick) plate modelp. 243
Kirchhoff (thin) plate modelp. 246
Boundary conditionsp. 248
Weak formulation and unique solvabilityp. 250
Babuska's paradox of thin platesp. 254
Discretization by H[superscript 2]-conforming elementsp. 255
Lowest-order (quintic) Argyris element, unisolvencyp. 255
Local interpolant, conformityp. 256
Nodal shape functions on the reference domainp. 257
Transformation to reference domainsp. 259
Design of basis functionsp. 260
Higher-order nodal Argyris-Fekete elementsp. 265
Exercisesp. 266
Equations of Electromagneticsp. 269
Electromagnetic field and its basic characteristicsp. 270
Integration along smooth curvesp. 270
Maxwell's equations in integral formp. 272
Maxwell's equations in differential formp. 273
Constitutive relations and the equation of continuityp. 274
Media and their characteristicsp. 275
Conductors and dielectricsp. 275
Magnetic materialsp. 276
Conditions on interfacesp. 277
Potentialsp. 279
Scalar electric potentialp. 279
Scalar magnetic potentialp. 281
Vector potential and gauge transformationsp. 281
Potential formulation of Maxwell's equationsp. 283
Other wave equationsp. 283
Equations for the field vectorsp. 284
Equation for the electric fieldp. 285
Equation for the magnetic fieldp. 285
Interface and boundary conditionsp. 286
Time-harmonic Maxwell's equationsp. 287
Helmholtz equationp. 288
Time-harmonic Maxwell's equationsp. 289
Normalizationp. 289
Model problemp. 290
Weak formulationp. 290
Existence and uniqueness of solutionp. 293
Edge elementsp. 300
Conformity requirements of the space H (curl)p. 301
Lowest-order (Whitney) edge elementsp. 302
Higher-order edge elements of Nedelecp. 309
Transformation of weak forms to the reference domainp. 314
Interpolation on edge elementsp. 316
Conformity of edge elements to the space H (curl)p. 317
Exercisesp. 318
Basics of Functional Analysisp. 319
Linear spacesp. 320
Real and complex linear spacep. 320
Checking whether a set is a linear spacep. 321
Intersection and union of subspacesp. 323
Linear combination and linear spanp. 326
Sum and direct sum of subspacesp. 327
Linear independence, basis, and dimensionp. 328
Linear operator, null space, rangep. 332
Composed operators and change of basisp. 337
Determinants, eigenvalues, and eigenvectorsp. 339
Hermitian, symmetric, and diagonalizable matricesp. 341
Linear forms, dual space, and dual basisp. 343
Exercisesp. 345
Normed spacesp. 348
Norm and seminormp. 348
Convergence and limitp. 352
Open and closed setsp. 355
Continuity of operatorsp. 357
Operator norm and [Gamma](U, V) as a normed spacep. 361
Equivalence of normsp. 363
Banach spacesp. 366
Banach fixed point theoremp. 371
Lebesgue integral and L[superscript p]-spacesp. 375
Basic inequalities in L[superscript p]-spacesp. 380
Density of smooth functions in L[superscript p]-spacesp. 384
Exercisesp. 386
Inner product spacesp. 389
Inner productp. 389
Hilbert spacesp. 394
Generalized angle and orthogonalityp. 395
Generalized Fourier seriesp. 399
Projections and orthogonal projectionsp. 401
Representation of linear forms (Riesz)p. 405
Compactness, compact operators, and the Fredholm alternativep. 407
Weak convergencep. 408
Exercisesp. 409
Sobolev spacesp. 412
Domain boundary and its regularityp. 412
Distributions and weak derivativesp. 414
Spaces W[superscript k,p] and H[superscript k]p. 418
Discontinuity of H[superscript 1]-functions in R[superscript d], d [greater than or equal] 2p. 420
Poincare-Friedrichs' inequalityp. 421
Embeddings of Sobolev spacesp. 422
Traces of W[superscript k,p]-functionsp. 424
Generalized integration by parts formulaep. 425
Exercisesp. 426
Software and Examplesp. 427
Sparse Matrix Solversp. 427
The sMatrix utilityp. 428
An example applicationp. 430
Interfacing with PETScp. 433
Interfacing with Trilinosp. 436
Interfacing with UMFPACKp. 439
The High-Performance Modular Finite Element System HERMESp. 439
Modular structure of HERMESp. 440
The elliptic modulep. 441
The Maxwell's modulep. 442
Example 1: L-shape domain problemp. 444
Example 2: Insulator problemp. 448
Example 3: Sphere-cone problemp. 451
Example 4: Electrostatic micromotor problemp. 455
Example 5: Diffraction problemp. 458
Referencesp. 461
Indexp. 468
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