List of Figures | p. xv |
List of Tables | p. xxi |
Preface | p. xxiii |
Acknowledgments | p. xxv |
Partial Differential Equations | p. 1 |
Selected general properties | p. 2 |
Classification and examples | p. 2 |
Hadamard's well-posedness | p. 5 |
General existence and uniqueness results | p. 9 |
Exercises | p. 11 |
Second-order elliptic problems | p. 13 |
Weak formulation of a model problem | p. 13 |
Bilinear forms, energy norm, and energetic inner product | p. 16 |
The Lax-Milgram lemma | p. 18 |
Unique solvability of the model problem | p. 18 |
Nonhomogeneous Dirichlet boundary conditions | p. 19 |
Neumann boundary conditions | p. 21 |
Newton (Robin) boundary conditions | p. 22 |
Combining essential and natural boundary conditions | p. 23 |
Energy of elliptic problems | p. 24 |
Maximum principles and well-posedness | p. 26 |
Exercises | p. 29 |
Second-order parabolic problems | p. 30 |
Initial and boundary conditions | p. 30 |
Weak formulation | p. 30 |
Existence and uniqueness of solution | p. 31 |
Exercises | p. 32 |
Second-order hyperbolic problems | p. 33 |
Initial and boundary conditions | p. 33 |
Weak formulation and unique solvability | p. 34 |
The wave equation | p. 34 |
Exercises | p. 35 |
First-order hyperbolic problems | p. 36 |
Conservation laws | p. 36 |
Characteristics | p. 38 |
Exact solution to linear first-order systems | p. 39 |
Riemann problem | p. 41 |
Nonlinear flux and shock formation | p. 43 |
Exercises | p. 44 |
Continuous Elements for 1D Problems | p. 45 |
The general framework | p. 45 |
The Galerkin method | p. 46 |
Orthogonality of error and Cea's lemma | p. 49 |
Convergence of the Galerkin method | p. 50 |
Ritz method for symmetric problems | p. 51 |
Exercises | p. 51 |
Lowest-order elements | p. 51 |
Model problem | p. 52 |
Finite-dimensional subspace V[subscript n subset or is implied by] V | p. 52 |
Piecewise-affine basis functions | p. 53 |
The system of linear algebraic equations | p. 54 |
Element-by-element assembling procedure | p. 55 |
Refinement and convergence | p. 56 |
Exercises | p. 57 |
Higher-order numerical quadrature | p. 59 |
Gaussian quadrature rules | p. 59 |
Selected quadrature constants | p. 61 |
Adaptive quadrature | p. 63 |
Exercises | p. 65 |
Higher-order elements | p. 66 |
Motivation problem | p. 66 |
Affine concept: reference domain and reference maps | p. 67 |
Transformation of weak forms to the reference domain | p. 69 |
Higher-order Lagrange nodal shape functions | p. 70 |
Chebyshev and Gauss-Lobatto nodal points | p. 71 |
Higher-order Lobatto hierarchic shape functions | p. 74 |
Constructing basis of the space V[subscript h,p] | p. 76 |
Data structures | p. 77 |
Assembling algorithm | p. 79 |
Exercises | p. 82 |
The sparse stiffness matrix | p. 84 |
Compressed sparse row (CSR) data format | p. 84 |
Condition number | p. 84 |
Conditioning of shape functions | p. 86 |
Stiffness matrix for the Lobatto shape functions | p. 88 |
Exercises | p. 89 |
Implementing nonhomogeneous boundary conditions | p. 89 |
Dirichlet boundary conditions | p. 89 |
Combination of essential and natural conditions | p. 91 |
Exercises | p. 92 |
Interpolation on finite elements | p. 93 |
The Hilbert space setting | p. 93 |
Best interpolant | p. 94 |
Projection-based interpolant | p. 96 |
Nodal interpolant | p. 99 |
Exercises | p. 102 |
General Concept of Nodal Elements | p. 103 |
The nodal finite element | p. 103 |
Unisolvency and nodal basis | p. 104 |
Checking unisolvency | p. 106 |
Example: lowest-order Q[superscript 1]-and P[superscript 1]-elements | p. 107 |
Q[superscript 1]-element | p. 108 |
P[superscript 1]-element | p. 110 |
Invertibility of the quadrilateral reference map x[subscript K] | p. 113 |
Interpolation on nodal elements | p. 114 |
Local nodal interpolant | p. 115 |
Global interpolant and conformity | p. 116 |
Conformity to the Sobolev space H[superscript 1] | p. 119 |
Equivalence of nodal elements | p. 120 |
Exercises | p. 122 |
Continuous Elements for 2D Problems | p. 125 |
Lowest-order elements | p. 126 |
Model problem and its weak formulation | p. 126 |
Approximations and variational crimes | p. 127 |
Basis of the space V[subscript h,p] | p. 129 |
Transformation of weak forms to the reference domain | p. 131 |
Simplified evaluation of stiffness integrals | p. 133 |
Connectivity arrays | p. 134 |
Assembling algorithm for Q[superscript 1]/P[superscript 1]-elements | p. 135 |
Lagrange interpolation on Q[superscript 1]/P[superscript 1]-meshes | p. 137 |
Exercises | p. 137 |
Higher-order numerical quadrature in 2D | p. 139 |
Gaussian quadrature on quads | p. 139 |
Gaussian quadrature on triangles | p. 139 |
Higher-order nodal elements | p. 142 |
Product Gauss-Lobatto points | p. 142 |
Lagrange-Gauss-Lobatto Q[superscript p,r]-elements | p. 143 |
Lagrange interpolation and the Lebesgue constant | p. 148 |
The Fekete points | p. 149 |
Lagrange-Fekete P[superscript p]-elements | p. 152 |
Basis of the space V[subscript h,p] | p. 154 |
Data structures | p. 157 |
Connectivity arrays | p. 160 |
Assembling algorithm for Q[superscript p]/P[superscript p]-elements | p. 162 |
Lagrange interpolation on Q[superscript p]/P[superscript p]-meshes | p. 166 |
Exercises | p. 166 |
Transient Problems and ODE Solvers | p. 167 |
Method of lines | p. 168 |
Model problem | p. 168 |
Weak formulation | p. 168 |
The ODE system | p. 169 |
Construction of the initial vector | p. 170 |
Autonomous systems and phase flow | p. 171 |
Selected time integration schemes | p. 172 |
One-step methods, consistency and convergence | p. 173 |
Explicit and implicit Euler methods | p. 175 |
Stiffness | p. 177 |
Explicit higher-order RK schemes | p. 179 |
Embedded RK methods and adaptivity | p. 182 |
General (implicit) RK schemes | p. 184 |
Introduction to stability | p. 185 |
Autonomization of RK methods | p. 186 |
Stability of linear autonomous systems | p. 187 |
Stability functions and stability domains | p. 188 |
Stability functions for general RK methods | p. 191 |
Maximum consistency order of IRK methods | p. 193 |
A-stability and L-stability | p. 194 |
Higher-order IRK methods | p. 197 |
Collocation methods | p. 197 |
Gauss and Radau IRK methods | p. 200 |
Solution of nonlinear systems | p. 202 |
Exercises | p. 205 |
Beam and Plate Bending Problems | p. 209 |
Bending of elastic beams | p. 210 |
Euler-Bernoulli model | p. 210 |
Boundary conditions | p. 212 |
Weak formulation | p. 214 |
Existence and uniqueness of solution | p. 214 |
Lowest-order Hermite elements in 1D | p. 216 |
Model problem | p. 216 |
Cubic Hermite elements | p. 218 |
Higher-order Hermite elements in 1D | p. 220 |
Nodal higher-order elements | p. 220 |
Hierarchic higher-order elements | p. 222 |
Conditioning of shape functions | p. 225 |
Basis of the space V[subscript h,p] | p. 226 |
Transformation of weak forms to the reference domain | p. 228 |
Connectivity arrays | p. 228 |
Assembling algorithm | p. 231 |
Interpolation on Hermite elements | p. 233 |
Hermite elements in 2D | p. 236 |
Lowest-order elements | p. 236 |
Higher-order Hermite-Fekete elements | p. 238 |
Design of basis functions | p. 240 |
Global nodal interpolant and conformity | p. 242 |
Bending of elastic plates | p. 242 |
Reissner-Mindlin (thick) plate model | p. 243 |
Kirchhoff (thin) plate model | p. 246 |
Boundary conditions | p. 248 |
Weak formulation and unique solvability | p. 250 |
Babuska's paradox of thin plates | p. 254 |
Discretization by H[superscript 2]-conforming elements | p. 255 |
Lowest-order (quintic) Argyris element, unisolvency | p. 255 |
Local interpolant, conformity | p. 256 |
Nodal shape functions on the reference domain | p. 257 |
Transformation to reference domains | p. 259 |
Design of basis functions | p. 260 |
Higher-order nodal Argyris-Fekete elements | p. 265 |
Exercises | p. 266 |
Equations of Electromagnetics | p. 269 |
Electromagnetic field and its basic characteristics | p. 270 |
Integration along smooth curves | p. 270 |
Maxwell's equations in integral form | p. 272 |
Maxwell's equations in differential form | p. 273 |
Constitutive relations and the equation of continuity | p. 274 |
Media and their characteristics | p. 275 |
Conductors and dielectrics | p. 275 |
Magnetic materials | p. 276 |
Conditions on interfaces | p. 277 |
Potentials | p. 279 |
Scalar electric potential | p. 279 |
Scalar magnetic potential | p. 281 |
Vector potential and gauge transformations | p. 281 |
Potential formulation of Maxwell's equations | p. 283 |
Other wave equations | p. 283 |
Equations for the field vectors | p. 284 |
Equation for the electric field | p. 285 |
Equation for the magnetic field | p. 285 |
Interface and boundary conditions | p. 286 |
Time-harmonic Maxwell's equations | p. 287 |
Helmholtz equation | p. 288 |
Time-harmonic Maxwell's equations | p. 289 |
Normalization | p. 289 |
Model problem | p. 290 |
Weak formulation | p. 290 |
Existence and uniqueness of solution | p. 293 |
Edge elements | p. 300 |
Conformity requirements of the space H (curl) | p. 301 |
Lowest-order (Whitney) edge elements | p. 302 |
Higher-order edge elements of Nedelec | p. 309 |
Transformation of weak forms to the reference domain | p. 314 |
Interpolation on edge elements | p. 316 |
Conformity of edge elements to the space H (curl) | p. 317 |
Exercises | p. 318 |
Basics of Functional Analysis | p. 319 |
Linear spaces | p. 320 |
Real and complex linear space | p. 320 |
Checking whether a set is a linear space | p. 321 |
Intersection and union of subspaces | p. 323 |
Linear combination and linear span | p. 326 |
Sum and direct sum of subspaces | p. 327 |
Linear independence, basis, and dimension | p. 328 |
Linear operator, null space, range | p. 332 |
Composed operators and change of basis | p. 337 |
Determinants, eigenvalues, and eigenvectors | p. 339 |
Hermitian, symmetric, and diagonalizable matrices | p. 341 |
Linear forms, dual space, and dual basis | p. 343 |
Exercises | p. 345 |
Normed spaces | p. 348 |
Norm and seminorm | p. 348 |
Convergence and limit | p. 352 |
Open and closed sets | p. 355 |
Continuity of operators | p. 357 |
Operator norm and [Gamma](U, V) as a normed space | p. 361 |
Equivalence of norms | p. 363 |
Banach spaces | p. 366 |
Banach fixed point theorem | p. 371 |
Lebesgue integral and L[superscript p]-spaces | p. 375 |
Basic inequalities in L[superscript p]-spaces | p. 380 |
Density of smooth functions in L[superscript p]-spaces | p. 384 |
Exercises | p. 386 |
Inner product spaces | p. 389 |
Inner product | p. 389 |
Hilbert spaces | p. 394 |
Generalized angle and orthogonality | p. 395 |
Generalized Fourier series | p. 399 |
Projections and orthogonal projections | p. 401 |
Representation of linear forms (Riesz) | p. 405 |
Compactness, compact operators, and the Fredholm alternative | p. 407 |
Weak convergence | p. 408 |
Exercises | p. 409 |
Sobolev spaces | p. 412 |
Domain boundary and its regularity | p. 412 |
Distributions and weak derivatives | p. 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] 2 | p. 420 |
Poincare-Friedrichs' inequality | p. 421 |
Embeddings of Sobolev spaces | p. 422 |
Traces of W[superscript k,p]-functions | p. 424 |
Generalized integration by parts formulae | p. 425 |
Exercises | p. 426 |
Software and Examples | p. 427 |
Sparse Matrix Solvers | p. 427 |
The sMatrix utility | p. 428 |
An example application | p. 430 |
Interfacing with PETSc | p. 433 |
Interfacing with Trilinos | p. 436 |
Interfacing with UMFPACK | p. 439 |
The High-Performance Modular Finite Element System HERMES | p. 439 |
Modular structure of HERMES | p. 440 |
The elliptic module | p. 441 |
The Maxwell's module | p. 442 |
Example 1: L-shape domain problem | p. 444 |
Example 2: Insulator problem | p. 448 |
Example 3: Sphere-cone problem | p. 451 |
Example 4: Electrostatic micromotor problem | p. 455 |
Example 5: Diffraction problem | p. 458 |
References | p. 461 |
Index | p. 468 |
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