## Spectral Methods: Evolution to Complex Geometries and Applications to Fluid DynamicsSpectral methods, particularly in their multidomain version, have become firmly established as a mainstream tool for scientific and engineering computation. While retaining the tight integration between the theoretical and practical aspects of spectral methods that was the hallmark of their 1988 book, Canuto et al. now incorporate the many improvements in the algorithms and the theory of spectral methods that have been made since then. This second new treatment, Evolution to Complex Geometries and Applications to Fluid Dynamics, provides an extensive overview of the essential algorithmic and theoretical aspects of spectral methods for complex geometries, in addition to detailed discussions of spectral algorithms for fluid dynamics in simple and complex geometries. Modern strategies for constructing spectral approximations in complex domains, such as spectral elements, mortar elements, and discontinuous Galerkin methods, as well as patching collocation, are introduced, analyzed, and demonstrated by means of numerous numerical examples. Representative simulations from continuum mechanics are also shown. Efficient domain decomposition preconditioners (of both Schwarz and Schur type) that are amenable to parallel implementation are surveyed. The discussion of spectral algorithms for fluid dynamics in single domains focuses on proven algorithms for the boundary-layer equations, linear and nonlinear stability analyses, incompressible Navier-Stokes problems, and both inviscid and viscous compressible flows. An overview of the modern approach to computing incompressible flows in general geometries using high-order, spectral discretizations is also provided. The recent companion book Fundamentals in Single Domains discusses the fundamentals of the approximation of solutions to ordinary and partial differential equations on single domains by expansions in smooth, global basis functions. The essential concepts and formulas from this book are included in the current text for the reader’s convenience. |

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### Contents

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13 Compressible Fluid Dynamics Equations | 7 |

131 Compressible NavierStokes Equations | 8 |

132 Nondimensionalization | 12 |

133 NavierStokes Equations with Turbulence Models | 13 |

562 Stability and Convergence Analysis | 283 |

Proof of the Global Infsup Condition | 285 |

563 Numerical Results | 286 |

57 The Mortar Element Method MEM | 289 |

571 Formulation of MEM | 290 |

572 Algebraic Aspects of MEM | 294 |

573 Analysis of MEM | 296 |

574 Other Applications | 299 |

134 Compressible Euler Equations | 17 |

136 Compressible BoundaryLayer Equations | 19 |

137 Compressible Stokes Limit | 20 |

138 Low Mach Number Compressible Limit | 21 |

142 Incompressible NavierStokes Equations with Turbulence Models | 22 |

143 VorticityStreamfunction Equations | 25 |

144 VorticityVelocity Equations | 26 |

145 Incompressible BoundaryLayer Equations | 27 |

151 Incompressible Linear Stability | 29 |

152 Compressible Linear Stability | 31 |

16 Stability Equations for Nonparallel Flows | 36 |

SingleDomain Algorithms and Applications for Stability Analysis | 39 |

22 BoundaryLayer Flows | 41 |

222 Compressible BoundaryLayer Flows | 48 |

23 Linear Stability of Incompressible Parallel Flows | 52 |

232 Numerical Examples for Plane Poiseuille Flow | 57 |

233 Some Other Incompressible Linear Stability Problems | 61 |

24 Linear Stability of Compressible Parallel Flows | 64 |

25 Nonparallel Linear Stability | 69 |

252 TwoDimensional Global Stability Analysis | 71 |

26 Transient Growth Analysis | 72 |

27 Nonlinear Stability | 75 |

272 Secondary Instability Theory | 77 |

273 Nonlinear Parabolized Stability Equations | 81 |

SingleDomain Algorithms and Applications for Incompressible Flows | 83 |

32 Conservation Properties and TimeDiscretization | 86 |

The Rotation Form | 88 |

The SkewSymmetric Form | 90 |

Convection and Divergence Forms | 92 |

323 Coupled Methods | 93 |

324 Splitting Methods | 95 |

325 Other Integration Methods | 96 |

33 Homogeneous Flows | 98 |

332 Dealiasing Using Transform Methods | 99 |

333 Pseudospectral and Collocation Methods | 103 |

334 Rogallo Transformation for Homogeneous Turbulence | 106 |

335 LargeEddy Simulation of Isotropic Turbulence | 108 |

Stability Accuracy and Aliasing | 110 |

34 Flows with One Inhomogeneous Direction | 121 |

341 Coupled Methods | 123 |

KleiserSchumann Algorithm | 124 |

Normal VelocityNormal Vorticity Algorithms | 127 |

342 Galerkin Methods Using DivergenceFree Bases | 131 |

343 Splitting Methods | 133 |

ZangHussaini Algorithm | 135 |

344 Other Conﬁned Flows | 138 |

345 Unbounded Flows | 140 |

FreeShearLayer Flows | 142 |

Accuracy | 144 |

35 Flows with Multiple Inhomogeneous Directions | 147 |

351 The Choice of Spatial Discretization in a Cavity | 149 |

352 The Choice of Spatial Discretization on a Reference Domain | 157 |

36 Outﬂow Boundary Conditions | 159 |

362 Buffer Domains | 161 |

37 Analysis of Spectral Methods for Incompressible Flows | 162 |

371 Compatibility Conditions Between Velocity and Pressure | 165 |

The Infsup Condition | 168 |

General Theory | 169 |

373 Speciﬁc Applications | 172 |

Numerical Results | 177 |

Extensions | 178 |

374 The Infsup Condition and the Pressure Operator | 179 |

The KleiserSchumann Method | 183 |

4 SingleDomain Methods for Compressible Flows | 186 |

421 Characteristic Compatibility Conditions | 188 |

The Characteristic Compatibility Method CCM | 189 |

CCM for a General 1D System | 192 |

CCM for the Collocation Method | 193 |

CCM for a General Multidimensional System | 195 |

References and Outlook | 196 |

422 Boundary Treatment for Linear Systems in Weak Formulations | 197 |

423 Spectral Accuracy and Conservation | 199 |

424 Analysis of Spectral Methods for Symmetric Hyperbolic Systems | 200 |

43 Boundary Treatment for the Euler Equations | 203 |

44 HighFrequency Control | 208 |

45 Homogeneous Turbulence | 211 |

452 Representative Applications | 214 |

46 Smooth Inhomogeneous Flows | 218 |

462 NavierStokes Equations | 221 |

463 Numerical Example | 224 |

47 Shock Fitting | 226 |

48 Shock Capturing | 233 |

5 Discretization Strategies | 237 |

52 The Spectral Element Method SEM in 1D | 239 |

522 Construction of SEM Basis Functions | 241 |

523 SEMNI and its Collocation Interpretation | 243 |

53 SEM for Multidimensional Problems | 245 |

532 Construction of SEM Basis Functions | 247 |

533 SEM and SEMNI Formulations | 250 |

534 Algebraic Aspects of SEM and SEMNI | 252 |

535 FiniteElement Preconditioning of SEMNI Matrices | 253 |

54 Analysis of SEM and SEMNI Approximations | 257 |

A Priori Error Analysis for SEM | 258 |

A Priori Analysis for SEMNI | 260 |

A Posteriori Error Analysis | 261 |

542 Multidimensional Analysis | 263 |

A Priori Error Analysis | 264 |

A Posteriori Error Analysis | 265 |

543 Some Proofs | 268 |

55 Some Numerical Results for the SEMNI Approximations | 273 |

552 Eigenfunction Approximation | 275 |

56 SEM for Stokes and NavierStokes Equations | 278 |

561 SEM and SEMNI Formulations | 279 |

58 The Spectral Discontinuous Galerkin Method SDGM in 1D | 300 |

581 Linear Advection Problems in ID | 301 |

582 Linear Hyperbolic Systems in 1D | 303 |

583 TimeDependent Problems | 308 |

584 Nonlinear Conservation Laws in 1D | 313 |

59 SDGM for Multidimensional Problems | 316 |

591 Multidimensional Formulation | 317 |

Nonlinear Conservation Laws | 319 |

592 The Mortar Technique for Geometrical Nonconformities | 321 |

510 SDGM for Diffusion Equations | 323 |

511 Analysis of SDGM | 326 |

512 SDGM for Euler and NavierStokes Equations | 332 |

TimeDiscretizations | 333 |

Numerical Examples | 334 |

Shock Tracking | 337 |

513 The Patching Method | 339 |

5132 Comparison of Patching and SEMNI | 343 |

5133 Collocation Methods for the Euler Equations | 345 |

Collocation Using a Nonstaggered Grid | 346 |

Collocation Using a Staggered Grid | 348 |

Multidomain Shock Fitting | 350 |

514 3D Applications in Complex Geometries | 352 |

Application to Compressible Flow | 353 |

Application to Thermoelasticity | 354 |

Structural Dynamics Analysis of the Roman Colosseum | 356 |

6 Solution Strategies for Spectral Methods in Complex Domains | 358 |

63 Overlapping Schwarz Alternating Methods | 364 |

631 Algebraic Form of Schwarz Methods for FiniteElement Discretization | 367 |

632 The Schwarz Method as an Algebraic Preconditioner | 370 |

633 Additive Schwarz Preconditioners for HighOrder Methods | 373 |

634 FEMSEM Spectral Equivalence | 380 |

635 Analysis of Schwarz Methods | 381 |

636 A General Theoretical Framework for the Analysis of DD Iterations | 383 |

64 Schur Complement Iterative Methods | 385 |

642 Properties of the SteklovPoincareOperator | 387 |

644 DD Preconditioners for the Schur Complement Matrix | 393 |

645 Preconditioners for the Stiffness Matrix Derived from Preconditioners for the Schur Complement Matrix | 398 |

65 Solution Algorithms for Patching Collocation Methods | 402 |

7 General Algorithms for Incompressible NavierStokes Equations | 407 |

72 HighOrder FractionalStep Methods | 409 |

73 Solution of the Algebraic System Associated with the Generalized Stokes Problem | 415 |

731 Preconditioners for the Generalized Stokes Matrix A | 416 |

732 Conditioning and Preconditioning for the Pressure Schur Complement Matrix | 419 |

733 Domain Decomposition Preconditioners for the Stokes and NavierStokes Equations | 421 |

74 Algebraic Factorization Methods | 425 |

742 Numerical Results for Yosida Schemes | 428 |

743 Preconditioners for the Approximate Pressure Schur Complement | 430 |

8 Spectral Methods Primer | 435 |

The Fourier Series | 436 |

Decay of the Fourier Coefficients | 437 |

Discrete Fourier Expansion and Interpolation | 438 |

Aliasing | 440 |

Differentiation | 441 |

Gibbs Phenomenon and Filtering | 443 |

82 General Jacobi Polynomials in the Interval 11 | 445 |

The Jacobi Series Truncation and Projection | 447 |

GaussType Quadrature Formulas and Discrete Inner Products | 448 |

Discrete Polynomial Transform and Interpolation | 449 |

Differentiation | 451 |

Quadrature Formulas and Discrete Transforms | 453 |

Differentiation | 454 |

84 Legendre Polynomials | 455 |

Quadrature Formulas and Discrete Norms | 457 |

85 Modal and Nodal BoundaryAdapted Bases on the Interval | 458 |

86 Orthogonal Systems in Unbounded Domains | 460 |

Hermite Polynomials and Hermite Functions | 461 |

87 Multidimensional Expansions | 462 |

871 TensorProduct Expansions | 463 |

872 Expansions on Simplicial Domains | 465 |

88 Mappings | 468 |

881 Finite Intervals | 469 |

882 SemiInﬁnite Intervals | 471 |

883 The Real Line | 473 |

884 Multidimensional Mappings on Finite Domains | 475 |

89 Basic Spectral Discretization Methods | 478 |

891 Tau Method | 479 |

892 Collocation Method | 481 |

893 Galerkin Method | 482 |

894 Galerkin with Numerical Integration GNI Method | 484 |

895 Other Boundary Conditions | 485 |

Appendix A Basic Mathematical Concepts | 488 |

A2 The CauchySchwarz Inequality | 491 |

A3 The LaxMilgram Theorem | 492 |

A5 The Spaces Cm𝜴 m 0 | 493 |

A7 Infinitely Differentiable Functions and Distributions | 494 |

A8 Sobolev Spaces and Sobolev Norms | 496 |

A9 The Sobolev Inequality | 501 |

Appendix B Fast Fourier Transforms | 503 |

Appendix C Iterative Methods for Linear Systems | 509 |

C2 Descent Methods for Symmetric Problems | 513 |

C3 Krylov Methods for Nonsymmetric Problems | 518 |

Appendix D Time Discretizations | 525 |

D2 Standard ODE Methods | 528 |

D21 Leap Frog Method | 529 |

D23 AdamsMoulton Methods | 531 |

D24 BackwardsDifference Formulas | 533 |

D25 RungeKutta Methods | 534 |

D3 LowStorage Schemes | 535 |

Appendix E Supplementary Material | 537 |

E2 Tau Correction for the KleiserSchumann Method | 539 |

E3 The Piola Transform | 541 |

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