Spectral and High Order Methods for Partial Differential Equations

1 - Foreword [Seite 6]
2 - Contents [Seite 8]
3 - hp-FEM for the Contact Problem with Tresca Friction in Linear Elasticity: The Primal Formulation [Seite 13]
3.1 - 1 Introduction [Seite 13]
3.2 - 2 Problem Formulation [Seite 14]
3.3 - 3 A Priori Error Estimates [Seite 16]
3.3.1 - 3.1 An Interpolation Error Estimate for B1/22,1-Functions [Seite 16]
3.3.2 - 3.2 A Polynomial Inverse Estimate [Seite 19]
3.3.3 - 3.3 Convergence Rates: Proof of Theorem 3.1 [Seite 21]
3.4 - 4 Numerical Experiments [Seite 25]
3.4.1 - 4.1 A Posteriori Error Estimation [Seite 25]
3.4.2 - 4.2 Numerical Examples [Seite 26]
3.5 - References [Seite 28]
4 - On Multivariate Chebyshev Polynomials and Spectral Approximations on Triangles [Seite 30]
4.1 - 1 Introduction [Seite 30]
4.2 - 2 Chebyshev Polynomials and Root Systems [Seite 33]
4.2.1 - 2.1 Root Systems [Seite 33]
4.2.2 - 2.2 Multivariate Chebyshev Polynomials [Seite 34]
4.2.3 - 2.3 The A2 Root System [Seite 36]
4.3 - 3 Computing Gradients [Seite 38]
4.3.1 - 3.1 Gradients in the A2 Root System [Seite 39]
4.4 - 4 Clenshaw-Curtis Quadrature [Seite 41]
4.4.1 - 4.1 Clenshaw-Curtis Quadrature in the A2 Root System [Seite 41]
4.5 - 5 Triangles [Seite 44]
4.5.1 - 5.1 Clenshaw-Curtis Quadrature Over a Triangle [Seite 44]
4.5.2 - 5.2 Nonlinear Transformations [Seite 46]
4.5.3 - 5.3 Linear Transformations [Seite 47]
4.6 - 6 Numerics [Seite 47]
4.7 - 7 Summary [Seite 50]
4.8 - References [Seite 51]
5 - Stochastic Spectral Galerkin and Collocation Methods for PDEs with Random Coefficients: A Numerical Comparison [Seite 53]
5.1 - 1 Introduction [Seite 53]
5.2 - 2 Problem Setting [Seite 55]
5.2.1 - 2.1 Finite Element Approximation in the Physical Space [Seite 57]
5.3 - 3 Polynomial Approximation in the Stochastic Dimension [Seite 57]
5.3.1 - 3.1 Stochastic Galerkin Approximation [Seite 59]
5.3.2 - 3.2 Stochastic Collocation Approximation on Sparse Grids [Seite 61]
5.4 - 4 Numerical Results [Seite 64]
5.4.1 - 4.1 Test Case 1: Isotropic Problem [Seite 64]
5.4.2 - 4.2 Test Case 2: Anisotropic Problem [Seite 68]
5.5 - References [Seite 71]
6 - Hybridizable Discontinuous Galerkin Methods [Seite 73]
6.1 - 1 Background [Seite 73]
6.2 - 2 The HDG Method [Seite 75]
6.2.1 - 2.1 The Convection-Diffusion Model Equation [Seite 75]
6.2.2 - 2.2 Mesh and Trace Operators [Seite 76]
6.2.3 - 2.3 Approximation Spaces [Seite 76]
6.2.4 - 2.4 HDG Formulation [Seite 77]
6.2.5 - 2.5 Characterization of the Numerical Trace [Seite 78]
6.2.6 - 2.6 Relation to Other DG Methods [Seite 79]
6.2.7 - 2.7 The Local Stabilization Parameter [Seite 81]
6.2.8 - 2.8 Local Postprocessing [Seite 81]
6.3 - 3 Extensions of the Basic Algorithm [Seite 83]
6.3.1 - 3.1 Time-Dependent Convection-Diffusion Problems [Seite 83]
6.3.2 - 3.2 Nonlinear Convection-Diffusion Problems [Seite 84]
6.3.3 - 3.3 Stokes Flows [Seite 85]
6.3.4 - 3.4 Incompressible Navier-Stokes Equations [Seite 88]
6.4 - 4 Numerical Results [Seite 90]
6.5 - 5 Conclusions [Seite 92]
6.6 - References [Seite 93]
7 - Multivariate Modified Fourier Expansions [Seite 95]
7.1 - 1 Introduction [Seite 95]
7.2 - 2 The d-Variate Cube [Seite 98]
7.3 - 3 The Hyperbolic Cross [Seite 98]
7.4 - 4 Accelerating Convergence [Seite 99]
7.4.1 - 4.1 The Lanczos Representation and Its Computation [Seite 99]
7.4.2 - 4.2 The Fourier Extension Problem [Seite 100]
7.5 - 5 The Non-Tensor Product Case [Seite 101]
7.6 - References [Seite 102]
8 - Constraint Oriented Spectral Element Method [Seite 103]
8.1 - 1 Introduction [Seite 103]
8.2 - 2 Constraint Oriented Polynomial Approximation [Seite 104]
8.2.1 - 2.1 Definition and Properties [Seite 104]
8.2.1.1 - 2.1.1 First Numerical Result [Seite 106]
8.2.2 - 2.2 Extension to Multidimensional Case [Seite 107]
8.3 - 3 The Constraint Oriented Effect [Seite 108]
8.3.1 - 3.1 Numerical Results [Seite 108]
8.4 - References [Seite 110]
9 - Convergence Rates of Sparse Tensor GPC FEM for Elliptic sPDEs [Seite 111]
9.1 - 1 Introduction [Seite 111]
9.2 - 2 Parametrization of the Model Problem [Seite 112]
9.2.1 - 2.1 Separation of Stochastic and Deterministic Variables [Seite 112]
9.2.2 - 2.2 Parametric Deterministic Problem [Seite 113]
9.3 - 3 Sparse Tensor Stochastic Galerkin Method [Seite 114]
9.3.1 - 3.1 Sparse Tensor Galerkin Formulation [Seite 114]
9.3.2 - 3.2 Hierarchic Discretization in L2() [Seite 115]
9.3.3 - 3.3 Hierarchic Discretization in D [Seite 116]
9.3.4 - 3.4 Convergence Rates of Sparse Tensor sGFEM [Seite 117]
9.4 - 4 Implementation and Numerical Examples [Seite 118]
9.4.1 - 4.1 Localization of Quasi-Best-N-Term Coefficients [Seite 118]
9.4.2 - 4.2 Numerical Example [Seite 118]
9.5 - References [Seite 119]
10 - A Conservative Spectral Element Method for Curvilinear Domains [Seite 121]
10.1 - 1 Introduction [Seite 121]
10.2 - 2 The Poisson Equation in Terms of Differential Forms [Seite 122]
10.3 - 3 Discretization of the Transformed Poisson Equation [Seite 123]
10.4 - 4 Results [Seite 127]
10.5 - 5 Concluding Remarks [Seite 128]
10.6 - References [Seite 128]
11 - An Efficient Control Variate Method for Parametrized Expectations [Seite 130]
11.1 - 1 A Control Variate Method for Parametrized Expectations [Seite 131]
11.1.1 - 1.1 Setting of the Problem [Seite 131]
11.1.2 - 1.2 The Control Variate Method [Seite 132]
11.1.3 - 1.3 A Practical Approach of the Control Variate Method Deduced from Parallels with the Standard Reduced-Basis Method [Seite 133]
11.2 - 2 Open Questions [Seite 136]
11.2.1 - 2.1 Rigorous Certification of the Variance Reduction? [Seite 136]
11.2.2 - 2.2 Computational Efficiency: Optimize MC Estimations? [Seite 137]
11.3 - References [Seite 139]
12 - A Proof, Based on the Euler Sum Acceleration, of the Recovery of an Exponential (Geometric) Rate of Convergence for the Fourier Series of a Function with Gibbs Phenomenon [Seite 140]
12.1 - 1 Introduction [Seite 140]
12.2 - 2 Acceleration by Conformal Mapping [Seite 142]
12.2.1 - 2.1 Abel Extension and Conformal Mapping [Seite 142]
12.2.2 - 2.2 Möbius Transformation and Euler Acceleration [Seite 143]
12.3 - 3 Accelerating a Fourier Series [Seite 144]
12.4 - 4 Numerical Illustration of Geometric Convergence [Seite 146]
12.5 - 5 Summary [Seite 147]
12.6 - References [Seite 147]
13 - A Seamless Reduced Basis Element Method for 2D Maxwell's Problem: An Introduction [Seite 149]
13.1 - 1 Introduction [Seite 150]
13.2 - 2 Reduced Basis Element Method [Seite 151]
13.2.1 - 2.1 Reduced Basis Method with Geometry As a Parameter [Seite 151]
13.2.2 - 2.2 Reduced Basis Element Method: Formulation [Seite 155]
13.2.3 - 2.3 Reduced Basis Element Method: Error Estimate [Seite 156]
13.3 - 3 Numerical Results [Seite 156]
13.3.1 - 3.1 Two-Parameter Case [Seite 157]
13.3.2 - 3.2 Three-Parameter Case [Seite 157]
13.4 - 4 Concluding Remarks [Seite 159]
13.5 - References [Seite 159]
14 - An hp-Nitsche's Method for Interface Problems with Nonconforming Unstructured Finite Element Meshes [Seite 161]
14.1 - 1 Introduction [Seite 161]
14.2 - 2 Discretization and Notations [Seite 162]
14.3 - 3 hp-Nitsche's Method [Seite 164]
14.4 - 4 Quasi-Optimal Convergence [Seite 166]
14.5 - References [Seite 169]
15 - Hybrid Explicit-Implicit Time Integration for Grid-Induced Stiffness in a DGTD Method for Time Domain Electromagnetics [Seite 170]
15.1 - 1 Introduction [Seite 170]
15.2 - 2 Continuous Problem [Seite 171]
15.3 - 3 Discretization in Space [Seite 172]
15.4 - 4 Time Discretization [Seite 172]
15.4.1 - 4.1 Explicit and Implicit Time Schemes [Seite 173]
15.4.2 - 4.2 Hybrid Explicit-Implicit Time Scheme [Seite 174]
15.5 - 5 Numerical Results [Seite 175]
15.6 - 6 Conclusions [Seite 177]
15.7 - References [Seite 177]
16 - High-Order Quasi-Uniform Approximation on the Sphere Using Fourier-Finite-Elements [Seite 178]
16.1 - 1 Introduction [Seite 178]
16.2 - 2 Quasi-Uniform Approximation of Scalar Fields by Fourier-Finite Elements [Seite 179]
16.3 - 3 Rotating Shallow-Water Equations [Seite 182]
16.4 - 4 Discussion [Seite 184]
16.5 - References [Seite 185]
17 - An hp Certified Reduced Basis Method for Parametrized Parabolic Partial Differential Equations [Seite 186]
17.1 - 1 Introduction [Seite 186]
17.2 - 2 The hp Reduced Basis Method [Seite 188]
17.3 - 3 A Convection-Diffusion Model Problem [Seite 191]
17.4 - References [Seite 193]
18 - Highly Accurate Discretization of the Navier-Stokes Equations in Streamfunction Formulation [Seite 195]
18.1 - 1 Fourth Order Scheme for the Navier-Stokes Equations in Two Dimensions [Seite 195]
18.2 - 2 The Pure Streamfunction Formulation in Three Dimensions [Seite 197]
18.3 - 3 The Numerical Scheme [Seite 199]
18.4 - References [Seite 202]
19 - Edge Functions for Spectral Element Methods [Seite 204]
19.1 - 1 Introduction [Seite 204]
19.2 - 2 The Edge Functions [Seite 205]
19.3 - 3 Application of Edge Functions to grad, curl and div [Seite 209]
19.4 - 4 Transformations [Seite 210]
19.5 - 5 Concluding Remarks [Seite 211]
19.6 - References [Seite 212]
20 - Modeling Effects of Electromagnetic Waves on Thin Wires with a High-Order Discontinuous Galerkin Method [Seite 213]
20.1 - 1 Introduction [Seite 213]
20.2 - 2 DG-FEM Discretization of Maxwell's Equations [Seite 214]
20.3 - 3 Thin Wire Equations and DG-FEM Discretization [Seite 215]
20.4 - 4 Field to Wire Coupling [Seite 216]
20.5 - 5 Wire to Field Coupling [Seite 217]
20.6 - 6 Full Field to Wire coupling [Seite 219]
20.7 - 7 Conclusion and Outlook [Seite 221]
20.8 - References [Seite 221]
21 - A Hybrid Method for the Resolution of the Gibbs Phenomenon [Seite 223]
21.1 - 1 Introduction [Seite 223]
21.2 - 2 The Inverse and Statistical Filter Methods [Seite 224]
21.2.1 - 2.1 Inverse Polynomial Reconstruction Method [Seite 224]
21.2.2 - 2.2 Statistical Filter Method [Seite 225]
21.3 - 3 Convergence, Accuracy and Exactness [Seite 226]
21.3.1 - 3.1 Convergence [Seite 226]
21.3.2 - 3.2 Covariance Matrix [Seite 227]
21.3.3 - 3.3 Spectral Accuracy and Exactness [Seite 228]
21.3.4 - 3.4 Numerical Convergence with Round-Off Errors [Seite 228]
21.4 - 4 A Hybrid IPRM and SF Method: Numerical Results [Seite 229]
21.5 - 5 Conclusions [Seite 230]
21.6 - References [Seite 230]
22 - Numerical Simulation of Fluid-Structure Interaction in Human Phonation: Verification of Structure Part [Seite 232]
22.1 - 1 Introduction [Seite 232]
22.2 - 2 Theory [Seite 233]
22.3 - 3 Summation by Parts Operators [Seite 234]
22.4 - 4 Application to Elastic Wave Equation [Seite 235]
22.5 - 5 Discretization [Seite 236]
22.6 - 6 Numerical Experiment [Seite 237]
22.7 - 7 Conclusions [Seite 239]
22.8 - References [Seite 239]
23 - A New Spectral Method on Triangles [Seite 240]
23.1 - 1 Introduction [Seite 240]
23.2 - 2 Rectangle-to-Triangle Mapping and Nodal Basis [Seite 241]
23.3 - 3 Implementations and Numerical Results [Seite 245]
23.4 - 4 Extensions and Discussions [Seite 247]
23.5 - References [Seite 248]
24 - The Reduced Basis Element Method: Offline-Online Decomposition in the Nonconforming, Nonaffine Case [Seite 250]
24.1 - 1 Introduction [Seite 250]
24.2 - 2 Offline-Online Decomposition [Seite 251]
24.3 - 3 A Posteriori Error Estimation [Seite 254]
24.4 - 4 Numerical Experiment [Seite 255]
24.5 - References [Seite 257]
25 - The Challenges of High Order Methods in Numerical Weather Prediction [Seite 258]
25.1 - 1 Introduction [Seite 258]
25.2 - 2 Overview of Atmospheric Modeling Challenges and Status [Seite 260]
25.3 - 3 Challenges [Seite 266]
25.3.1 - 3.1 Where High Order Holds Promise [Seite 266]
25.3.2 - 3.2 Where High Order Instills Doubts [Seite 266]
25.3.3 - 3.3 Recommendations [Seite 267]
25.4 - 4 Conclusions [Seite 267]
25.5 - References [Seite 268]
26 - GMRES for Oscillatory Matrix-Valued Differential Equations [Seite 270]
26.1 - 1 Introduction [Seite 270]
26.2 - 2 Oscillatory Integrals [Seite 272]
26.3 - 3 Oscillatory Differential Equations [Seite 274]
26.4 - 4 Example: Mathieu Functions [Seite 276]
26.5 - References [Seite 277]
27 - Sensitivity Analysis of Heat Exchangers Using Perturbative Methods [Seite 278]
27.1 - 1 Introduction [Seite 278]
27.2 - 2 The One-Dimensional Horizontal Heat Exchanger Problem [Seite 279]
27.3 - 3 Reference Case [Seite 281]
27.4 - 4 Sensitivity Analysis Results [Seite 282]
27.5 - 5 Sensitivity Analysis Accuracy [Seite 283]
27.6 - 6 Conclusions [Seite 284]
27.7 - References [Seite 285]
28 - Spectral Element Approximation of the Hodge- Operator in Curved Elements [Seite 286]
28.1 - 1 Introduction [Seite 286]
28.2 - 2 Mimetic Approaches for the 2D Poisson Equation [Seite 287]
28.3 - 3 Weak Material Laws: The Role of Least-Squares [Seite 288]
28.4 - 4 Application to the 2D Poisson Equation [Seite 289]
28.4.1 - 4.1 Straight Elements [Seite 289]
28.4.2 - 4.2 Curved Elements [Seite 289]
28.4.2.1 - 4.2.1 The Inner Product [Seite 290]
28.4.2.2 - 4.2.2 The Hodge- Operator [Seite 291]
28.4.2.3 - 4.2.3 The Least-Squares Residual [Seite 292]
28.5 - 5 Concluding Remarks [Seite 292]
28.6 - References [Seite 293]
29 - Uncertainty Propagation for Systems of Conservation Laws, High Order Stochastic Spectral Methods [Seite 295]
29.1 - 1 Mathematical Framework [Seite 296]
29.1.1 - 1.1 SLC in a Nutshell [Seite 296]
29.1.2 - 1.2 gPC in a Nutshell [Seite 296]
29.2 - 2 Application of sG-gPC to the p-System in Lagrangian Coordinates [Seite 297]
29.2.1 - 2.1 Closure of (6) or Treatment of Non Linearities [Seite 299]
29.2.2 - 2.2 Discontinuous Solutions and Gibbs Phenomenon [Seite 301]
29.3 - 3 The Intrusive Polynomial Moment Method (IPMM) [Seite 301]
29.3.1 - 3.1 Analogy with Kt and Mt for the Closure [Seite 302]
29.4 - 4 Numerical Tests [Seite 304]
29.4.1 - 4.1 Comparison Between sG-gPC and IPMM: Burgers [Seite 304]
29.4.2 - 4.2 Stochastic Riemann Problem: Euler System [Seite 305]
29.5 - 5 Conclusions [Seite 306]
29.6 - References [Seite 307]
30 - Reduced Basis Approximation for Shape Optimization in Thermal Flows with a Parametrized Polynomial Geometric Map [Seite 308]
30.1 - 1 Introduction [Seite 308]
30.2 - 2 Reduced Basis Approximation of Parametric Advection-Diffusion Equations [Seite 309]
30.3 - 3 Numerical Example [Seite 313]
30.4 - 4 Conclusions [Seite 314]
30.5 - References [Seite 315]
31 - Constrained Approximation in hp-FEM: Unsymmetric Subdivisions and Multi-Level Hanging Nodes [Seite 317]
31.1 - 1 Introduction [Seite 317]
31.2 - 2 Tensor Product Shape Functions of Legendre Type [Seite 318]
31.3 - 3 Constraints Coefficients and Multi-Level Hanging Nodes [Seite 321]
31.4 - 4 Numerical Results [Seite 323]
31.5 - References [Seite 324]
32 - High Order Filter Methods for Wide Range of Compressible Flow Speeds [Seite 326]
32.1 - 1 Original High Order Filter Method [Seite 326]
32.2 - 2 Improved High Order Filter Method [Seite 328]
32.3 - 3 Numerical Results [Seite 330]
32.3.1 - 3.1 1-D Shock/Turbulence Interaction Problem [Seite 331]
32.3.2 - 3.2 Taylor-Green Vortex [Seite 332]
32.3.3 - 3.3 Compressible Isotropic Turbulence with Shocklets [Seite 333]
32.4 - 4 New Flow Sensor for a Wide Spectrum of Flow Speedand Shock Strength [Seite 334]
32.5 - References [Seite 335]
33 - hp-Adaptive CEM in Practical Applications [Seite 337]
33.1 - 1 Introduction [Seite 337]
33.2 - 2 The Scattering Matrix Based Approach [Seite 338]
33.3 - 3 Results [Seite 340]
33.3.1 - 3.1 Fully Coupled Interior and Exterior Model [Seite 340]
33.3.2 - 3.2 Scattering Matrix Based Interior Only Model [Seite 342]
33.4 - References [Seite 344]
34 - Anchor Points Matter in ANOVA Decomposition [Seite 345]
34.1 - 1 Introduction [Seite 345]
34.2 - 2 Weights and Effective Dimension [Seite 346]
34.3 - 3 Numerical Examples [Seite 351]
34.4 - References [Seite 353]
35 - An Explicit Discontinuous Galerkin Scheme with Divergence Cleaning for Magnetohydrodynamics [Seite 354]
35.1 - 1 Introduction [Seite 354]
35.2 - 2 STE-DG Discretization [Seite 355]
35.2.1 - 2.1 Space-Time Expansion [Seite 355]
35.2.2 - 2.2 Local Time Stepping [Seite 356]
35.3 - 3 Divergence Correction and Local Time Stepping [Seite 356]
35.4 - 4 Numerical Results [Seite 359]
35.4.1 - 4.1 Convergence Test [Seite 359]
35.4.2 - 4.2 Orszag-Tang Vortex [Seite 359]
35.5 - 5 Conclusions [Seite 360]
35.6 - References [Seite 361]
36 - High Order Polynomial Interpolation of Parameterized Curves [Seite 362]
36.1 - 1 Introduction [Seite 362]
36.2 - 2 Interpolation Methods for Plane Curves [Seite 363]
36.2.1 - 2.1 Common Interpolation Methods [Seite 364]
36.2.2 - 2.2 The L2-Method [Seite 364]
36.2.3 - 2.3 The Equal-Tangent Method [Seite 365]
36.2.4 - 2.4 Numerical Results [Seite 365]
36.3 - 3 Interpolation of Space Curves [Seite 366]
36.3.1 - 3.1 The L2-Method [Seite 366]
36.3.2 - 3.2 The Equal-Tangent Method [Seite 367]
36.3.3 - 3.3 Numerical Results [Seite 367]
36.4 - 4 Conclusions and Future Work [Seite 368]
36.5 - References [Seite 369]
37 - A New Discontinuous Galerkin Method for the Navier-Stokes Equations [Seite 370]
37.1 - 1 Introduction [Seite 370]
37.2 - 2 Numerical Discretization [Seite 371]
37.2.1 - 2.1 DG Formulation and Time Stepping [Seite 371]
37.2.2 - 2.2 The Elastoplast Method (EDG) [Seite 372]
37.2.2.1 - 2.2.1 DG Basis and Implementation [Seite 373]
37.3 - 3 Numerical Results [Seite 373]
37.3.1 - 3.1 1D Diffusion [Seite 374]
37.3.2 - 3.2 Couette Thermal Flow [Seite 374]
37.3.3 - 3.3 Blasius Boundary Layer [Seite 375]
37.3.4 - 3.4 Supersonic Mixing Layer [Seite 376]
37.4 - 4 Conclusions [Seite 377]
37.5 - References [Seite 377]
38 - A Pn,-Based Method for Linear Nonconstant Coefficients High Order Eigenvalue Problems [Seite 379]
38.1 - 1 Introduction [Seite 379]
38.2 - 2 Physical and Mathematical Preliminaries [Seite 381]
38.3 - 3 Numerical Results [Seite 383]
38.3.1 - 3.1 The Rigid Boundaries Case [Seite 383]
38.3.2 - 3.2 The Free Boundary Case [Seite 384]
38.4 - 4 Conclusions [Seite 386]
38.5 - References [Seite 387]
39 - Spectral Element Discretization of Optimal Control Problems [Seite 388]
39.1 - 1 Linear Optimal Control Problem [Seite 389]
39.2 - 2 SEM Discretization [Seite 390]
39.3 - 3 Iteration and Discretization Error Estimates [Seite 392]
39.4 - 4 Numerical Results [Seite 394]
39.5 - 5 Conclusions [Seite 394]
39.6 - References [Seite 396]
40 - Applications of High Order Methods to Vortex Instability Calculations [Seite 397]
40.1 - 1 Introduction [Seite 397]
40.2 - 2 Theory [Seite 399]
40.2.1 - 2.1 The Basic Flows [Seite 399]
40.2.2 - 2.2 The BiGlobal Eigenvalue Problem (EVP) [Seite 400]
40.3 - 3 Results [Seite 401]
40.3.1 - 3.1 Basic Flow [Seite 401]
40.3.2 - 3.2 Instability Analyses [Seite 401]
40.4 - 4 Conclusions and Outlook [Seite 403]
40.5 - References [Seite 404]
41 - Entropy Viscosity Method for High-Order Approximations of Conservation Laws [Seite 405]
41.1 - 1 Introduction [Seite 405]
41.2 - 2 The Entropy Viscosity Method [Seite 406]
41.3 - 3 2D Burgers (Fourier) [Seite 408]
41.4 - 4 KPP Rotating Wave (SEM) [Seite 409]
41.5 - 5 2D Euler System (Fourier) [Seite 411]
41.6 - References [Seite 412]
42 - High-Order Accurate Numerical Solution of Incompressible Slip Flow and Heat Transfer in Microchannels [Seite 413]
42.1 - 1 Introduction [Seite 413]
42.2 - 2 Problem Formulation [Seite 414]
42.3 - 3 Wall Boundary Conditions [Seite 415]
42.4 - 4 Numerical Procedure [Seite 415]
42.5 - 5 Numerical Results and Discussion [Seite 416]
42.6 - 6 Concluding Remarks [Seite 420]
42.7 - References [Seite 421]
43 - Spectral Methods for Time-Dependent Variable-Coefficient PDE Based on Block Gaussian Quadrature [Seite 422]
43.1 - 1 Introduction [Seite 422]
43.2 - 2 Krylov Subspace Spectral Methods [Seite 423]
43.3 - 3 Implementation [Seite 426]
43.4 - 4 Application to Maxwell's Equations [Seite 426]
43.5 - 5 Numerical Results [Seite 428]
43.5.1 - 5.1 Parabolic Problems [Seite 428]
43.5.2 - 5.2 Maxwell's Equations [Seite 429]
43.6 - 6 Summary and Future Work [Seite 431]
43.7 - References [Seite 431]
44 - The Spectral Element Method Used to Assess the Quality of a Global C1 Map [Seite 433]
44.1 - 1 Introduction [Seite 433]
44.2 - 2 Methods [Seite 434]
44.3 - 3 Regularity [Seite 439]
44.4 - References [Seite 440]
45 - Stabilization of the Spectral-Element Method in Turbulent Flow Simulations [Seite 441]
45.1 - 1 Introduction [Seite 441]
45.2 - 2 Equations and Discretization [Seite 442]
45.3 - 3 Stabilization of Turbulent Flow Simulations [Seite 443]
45.4 - 4 Analysis of Model Problems [Seite 443]
45.4.1 - 4.1 1D: Stabilization of the Burgers' Equation [Seite 443]
45.4.2 - 4.2 2D: Recovery of Skew-Symmetry for the SEM Convection Operator in the Scalar Transport Equation [Seite 445]
45.5 - 5 Application to the Navier-Stokes Equations [Seite 446]
45.5.1 - 5.1 3D: Subcritical K-type Transition Simulations [Seite 447]
45.5.2 - 5.2 3D: Fully Turbulent Channel Flow Simulationsat Re = 590 [Seite 448]
45.6 - 6 Conclusions [Seite 449]
45.7 - References [Seite 449]
46 - The Spectral-Element and Pseudo-Spectral Methods: A Comparative Study [Seite 451]
46.1 - 1 Introduction [Seite 451]
46.2 - 2 Study Setup [Seite 452]
46.3 - 3 Results [Seite 453]
46.3.1 - 3.1 Part A: Efficiency [Seite 453]
46.3.2 - 3.2 Part B: Accuracy in Transitional Flow Simulations [Seite 454]
46.3.3 - 3.3 Part B: Accuracy in Turbulent Flow Simulations [Seite 455]
46.4 - 4 Conclusions [Seite 457]
46.5 - References [Seite 458]
47 - Adaptive Spectral Filtering and Digital Total Variation Postprocessing for the DG Method on Triangular Grids: Application to the Euler Equations [Seite 460]
47.1 - 1 Introduction [Seite 460]
47.2 - 2 The Discontinuous Galerkin Scheme with Spectral Filtering [Seite 461]
47.3 - 3 The Digital Total Variation Filter [Seite 462]
47.4 - 4 Numerical Experiments [Seite 463]
47.5 - References [Seite 467]
48 - BDDC and FETI-DP Preconditioners for Spectral Element Discretizations of Almost Incompressible Elasticity [Seite 469]
48.1 - 1 Introduction [Seite 469]
48.2 - 2 Almost Incompressible Elasticity and Spectral Elements [Seite 469]
48.3 - 3 The BDDC Algorithm [Seite 471]
48.4 - 4 Numerical Results in the Plane [Seite 474]
48.5 - References [Seite 475]
49 - A Two-Dimensional DG-SEM Approach to Investigate Resonance Frequencies and Sound Radiation of Woodwind Instruments [Seite 477]
49.1 - 1 Introduction [Seite 477]
49.2 - 2 Discontinuous Galerkin Method for the Euler Equations [Seite 479]
49.2.1 - 2.1 Conservation Equations [Seite 479]
49.2.2 - 2.2 Numerical Scheme [Seite 479]
49.3 - 3 The Influence of the Vocal Tract on the Recorder [Seite 480]
49.3.1 - 3.1 Problem Description [Seite 480]
49.3.2 - 3.2 Influence of the Vocal Tract [Seite 481]
49.4 - 4 Sound Radiation of the Bassoon [Seite 482]
49.5 - 5 Conclusions [Seite 484]
49.6 - References [Seite 484]
50 - Spectral Properties of Discontinuous Galerkin Space Operators on Curved Meshes [Seite 485]
50.1 - 1 Introduction [Seite 485]
50.2 - 2 Method [Seite 486]
50.2.1 - 2.1 Discontinuous Galerkin Method [Seite 486]
50.2.2 - 2.2 Stability Analysis [Seite 487]
50.3 - 3 Results [Seite 488]
50.3.1 - 3.1 Qualitative Results in 2D [Seite 489]
50.3.2 - 3.2 Dependence on the Local Jacobian in 1D [Seite 490]
50.3.3 - 3.3 Estimation Based on Integration Matrices in 1D [Seite 491]
50.4 - 4 Conclusions [Seite 491]
50.5 - References [Seite 492]
51 - Post-Processing of Marginally Resolved Spectral Element Data [Seite 493]
51.1 - 1 Introduction [Seite 493]
51.2 - 2 Numerical Test Problems [Seite 494]
51.2.1 - 2.1 An Analytical Example [Seite 494]
51.2.2 - 2.2 Turbulent Channel Flow [Seite 494]
51.3 - 3 Interface Averaging [Seite 495]
51.4 - 4 Improved Interface Treatment [Seite 497]
51.4.1 - 4.1 Polynomial Interpolation [Seite 497]
51.4.2 - 4.2 Filtering [Seite 499]
51.5 - 5 Conclusions [Seite 499]
51.6 - References [Seite 500]
52 - Editorial Policy [Seite 501]
53 - Lecture Notes in Computational Science and Engineering [Seite 503]
54 - Monographs in Computational Science and Engineering [Seite 507]
55 - Texts in Computational Science and Engineering [Seite 507]