Domain Decomposition Methods in Science and Engineering XIX
1st Edition
Published on 27. October 2010
XXIV, 472 pages
978-3-642-11304-8 (ISBN)
Description
These are the proceedings of the 19th international conference on domain decomposition methods in science and engineering. Domain decomposition methods are iterative methods for solving the often very large linear or nonlinear systems of algebraic equations that arise in various problems in mathematics, computational science, engineering and industry. They are designed for massively parallel computers and take the memory hierarchy of such systems into account. This is essential for approaching peak floating point performance. There is an increasingly well-developed theory which is having a direct impact on the development and improvement of these algorithms.
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Yunqing Huang | Ralf Kornhuber | Olof Widlund
Domain Decomposition Methods in Science and Engineering XIX
Book
10/2010
Springer
€106.99
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Persons
The editors are all well-known researchers in this research field.
Content
1 - Preface [Seite 5]
2 - Contents [Seite 10]
3 - Contributors [Seite 15]
4 - Part I Plenary Presentations [Seite 23]
4.1 - Domain Decomposition and hp-Adaptive Finite Elements [Seite 24]
4.1.1 - 1 Introduction [Seite 24]
4.1.2 - 2 A Posteriori Error Estimate [Seite 25]
4.1.3 - 3 Basis Functions [Seite 26]
4.1.4 - 4 Parallel Adaptive Algorithm [Seite 27]
4.1.5 - 5 DD Solver [Seite 28]
4.1.6 - 6 Numerical Results [Seite 31]
4.1.7 - Bibliography [Seite 34]
4.2 - Domain Decomposition Methods for Electromagnetic Wave Propagation Problems in Heterogeneous Media and Complex Domains [Seite 35]
4.2.1 - 1 Introduction [Seite 35]
4.2.2 - 2 Continuous Problem [Seite 36]
4.2.3 - 3 A Family of Schwarz DD Algorithms [Seite 37]
4.2.4 - 4 Discretization by a High Order DG Method [Seite 38]
4.2.4.1 - 4.1 Discretization of the Monodomain Problem [Seite 38]
4.2.4.2 - 4.2 Discretization of the DD Algorithm [Seite 39]
4.2.4.2.1 - DG Formulation of the Multi-Domain Problem [Seite 39]
4.2.4.2.2 - Formulation of an Interface System [Seite 40]
4.2.5 - 5 Numerical Results [Seite 41]
4.2.5.1 - 5.1 The 2D Case [Seite 41]
4.2.5.2 - 5.2 The 3D Case [Seite 43]
4.2.6 - 6 Ongoing and Future Work [Seite 44]
4.2.7 - Bibliography [Seite 45]
4.3 - N--N Solvers for a DG Discretization for Geometrically Nonconforming Substructures and Discontinuous Coefficients [Seite 47]
4.3.1 - 1 Summary [Seite 47]
4.3.2 - 2 Introduction [Seite 47]
4.3.3 - 3 Differential and Discrete Problems [Seite 49]
4.3.3.1 - 3.1 Differential Problem [Seite 49]
4.3.3.2 - 3.2 Discrete Problem [Seite 49]
4.3.3.3 - 3.3 Schur Complement Problem [Seite 50]
4.3.4 - 4 Notation and the Interface Condition [Seite 53]
4.3.5 - 5 Additive Preconditioners [Seite 55]
4.3.5.1 - 5.1 Local Problems [Seite 55]
4.3.5.2 - 5.2 Coarse Problems [Seite 56]
4.3.5.3 - 5.3 Condition Number Estimate for Tas,I [Seite 56]
4.3.6 - 6 Final Remarks [Seite 57]
4.3.7 - Bibliography [Seite 57]
4.4 - On Adaptive-Multilevel BDDC [Seite 59]
4.4.1 - 1 Introduction [Seite 59]
4.4.2 - 2 Abstract BDDC for a Model Problem [Seite 60]
4.4.2.1 - 2.1 Multilevel BDDC [Seite 61]
4.4.3 - 3 Indicator of the Condition Number Bound [Seite 63]
4.4.4 - 4 Optimal Coarse Degrees of Freedom [Seite 64]
4.4.5 - 5 Adaptive-Multilevel BDDC in 2D [Seite 65]
4.4.6 - 6 Numerical Examples and Conclusion [Seite 66]
4.4.7 - Bibliography [Seite 70]
4.5 - Interpolation Based Local Postprocessing for Adaptive Finite Element Approximations in Electronic Structure Calculations [Seite 71]
4.5.1 - 1 Introduction [Seite 71]
4.5.2 - 2 Interpolation Based Finite Element Postprocessing [Seite 73]
4.5.2.1 - 2.1 Finite Element Discretizations [Seite 74]
4.5.2.2 - 2.2 Interpolation Based Local Postprocessing [Seite 75]
4.5.2.3 - 2.3 Quantum Harmonic Oscillator [Seite 75]
4.5.3 - 3 Applications to Electronic Structure Calculations [Seite 76]
4.5.3.1 - 3.1 Linearization of Kohn--Sham Equation [Seite 76]
4.5.3.2 - 3.2 Experiments [Seite 77]
4.5.3.2.1 - Benzene [Seite 78]
4.5.3.2.2 - Fullerene [Seite 79]
4.5.4 - 4 Concluding Remarks [Seite 80]
4.5.5 - Bibliography [Seite 80]
4.6 - A New a Posteriori Error Estimate for Adaptive Finite Element Methods [Seite 82]
4.6.1 - 1 Introduction [Seite 82]
4.6.2 - 2 A Posteriori Error Estimate [Seite 83]
4.6.3 - 3 Numerical Validation and Applications [Seite 89]
4.6.4 - Bibliography [Seite 92]
4.7 - Space-Time Nonconforming Optimized Schwarz Waveform Relaxation for Heterogeneous Problems and General Geometries [Seite 94]
4.7.1 - 1 Introduction [Seite 94]
4.7.2 - 2 The Continuous OSWR Algorithm [Seite 95]
4.7.3 - 3 Numerical Results [Seite 100]
4.7.4 - 4 Conclusions [Seite 103]
4.7.5 - Bibliography [Seite 105]
4.8 - Convergence Behaviour of Dirichlet--Neumann and Robin Methods for a Nonlinear Transmission Problem [Seite 106]
4.8.1 - 1 Introduction [Seite 106]
4.8.2 - 2 Transmission Problem with Jumping Nonlinearities [Seite 108]
4.8.3 - 3 Nonlinear Dirichlet--Neumann and Robin Methods [Seite 109]
4.8.3.1 - 3.1 The Methods and Their Steklov--Poincaré Formulations [Seite 109]
4.8.3.2 - 3.2 Convergence Results [Seite 110]
4.8.4 - 4 Parameter Studies for the Dirichlet--Neumann Method [Seite 111]
4.8.5 - 5 Parameter Studies for the Robin Method [Seite 114]
4.8.6 - Bibliography [Seite 117]
5 - Part II Minisymposia [Seite 118]
5.1 - Optimal Interface Conditions for an Arbitrary Decomposition into Subdomains [Seite 119]
5.1.1 - 1 Optimal Interface Conditions [Seite 119]
5.1.2 - 2 Notation and Assumptions [Seite 120]
5.1.3 - 3 Construction of the Method [Seite 120]
5.1.4 - 4 Sparsity Pattern [Seite 122]
5.1.5 - 5 Numerical Examples [Seite 124]
5.1.6 - 6 Conclusion [Seite 125]
5.1.7 - Bibliography [Seite 126]
5.2 - Optimized Schwarz Methods for Domains with an Arbitrary Interface [Seite 127]
5.2.1 - 1 Introduction [Seite 127]
5.2.2 - 2 First-Order Boundary Condition [Seite 128]
5.2.3 - 3 Higher-Order Boundary Condition [Seite 130]
5.2.4 - Bibliography [Seite 133]
5.3 - Can the Discretization Modify the Performance of Schwarz Methods? [Seite 135]
5.3.1 - 1 Introduction [Seite 135]
5.3.2 - 2 The Cauchy--Riemann Equations [Seite 135]
5.3.3 - 3 The Positive Definite Helmholtz Equation [Seite 140]
5.3.4 - 4 Conclusions [Seite 141]
5.3.5 - Bibliography [Seite 141]
5.4 - The Pole Condition: A Padé Approximation of the Dirichlet to Neumann Operator [Seite 143]
5.4.1 - 1 Introduction [Seite 143]
5.4.2 - 2 Model Problem [Seite 144]
5.4.3 - 3 The Pole Condition [Seite 145]
5.4.4 - 4 Error Estimate [Seite 147]
5.4.5 - Bibliography [Seite 149]
5.5 - Discontinuous Galerkin and Nonconforming in Time Optimized Schwarz Waveform Relaxation [Seite 151]
5.5.1 - 1 Introduction [Seite 151]
5.5.2 - 2 Local Problem and Time Discontinuous Galerkin [Seite 152]
5.5.3 - 3 The Optimized Schwarz Waveform Relaxation Algorithm Discretized in Time with Different Subdomain Grids [Seite 153]
5.5.4 - 4 Numerical Results [Seite 156]
5.5.5 - 5 Conclusions [Seite 158]
5.5.6 - Bibliography [Seite 158]
5.6 - Two-Level Methods for Blood Flow Simulation [Seite 159]
5.6.1 - 1 Introduction [Seite 159]
5.6.2 - 2 Mathematical Model and Discretization [Seite 159]
5.6.3 - 3 Two-Level Newton and Schwarz Methods [Seite 161]
5.6.4 - 4 Numerical Results [Seite 163]
5.6.5 - 5 Conclusion [Seite 166]
5.6.6 - Bibliography [Seite 166]
5.7 - Newton-Krylov-Schwarz Method for a Spherical Shallow Water Model [Seite 167]
5.7.1 - 1 Introduction [Seite 167]
5.7.2 - 2 Governing Equations [Seite 167]
5.7.3 - 3 Discretizations [Seite 168]
5.7.4 - 4 Nonlinear Solver [Seite 169]
5.7.5 - 5 Numerical Results [Seite 170]
5.7.6 - Bibliography [Seite 172]
5.8 - A Parallel Scalable PETSc-Based Jacobi-Davidson Polynomial Eigensolver with Application in Quantum Dot Simulation [Seite 174]
5.8.1 - 1 Introduction [Seite 174]
5.8.2 - 2 A Description of the ASPJD Algorithm [Seite 175]
5.8.3 - 3 A PETSc-Based ASPJD Polynomial Eigensolver [Seite 177]
5.8.4 - 4 Numerical Results [Seite 178]
5.8.5 - Bibliography [Seite 180]
5.9 - Two-Level Multiplicative Domain Decomposition Algorithm for Recovering the Lamé Coefficient in Biological Tissues [Seite 182]
5.9.1 - 1 Introduction [Seite 182]
5.9.2 - 2 Recovering the Lamé Coefficient in Biological Tissues [Seite 182]
5.9.3 - 3 Lagrange-Newton-Krylov-Schwarz Algorithm [Seite 184]
5.9.4 - 4 Numerical Results and Discussion [Seite 186]
5.9.5 - 5 Concluding Remarks [Seite 188]
5.9.6 - Bibliography [Seite 189]
5.10 - Robust Preconditioner for H(curl) Interface Problems [Seite 190]
5.10.1 - 1 Introduction [Seite 190]
5.10.2 - 2 Regular Decomposition [Seite 191]
5.10.3 - 3 Auxiliary Space Preconditioners [Seite 194]
5.10.4 - 4 Conclusions [Seite 196]
5.10.5 - Bibliography [Seite 196]
5.11 - Mixed Multiscale Finite Element Analysis for Wave Equations Using Global Information [Seite 198]
5.11.1 - 1 Introduction [Seite 198]
5.11.2 - 2 Preliminaries [Seite 199]
5.11.3 - 3 Mixed MsFEM Analysis [Seite 200]
5.11.3.1 - 3.1 Mixed MsFEM Formulation [Seite 200]
5.11.3.2 - 3.2 A Priori Error Estimates for Continuous Time [Seite 202]
5.11.3.3 - 3.3 A Priori Error Estimate for Discrete Time [Seite 204]
5.11.4 - 4 Conclusions [Seite 205]
5.11.5 - Bibliography [Seite 205]
5.12 - A Domain Decomposition Preconditioner for Multiscale High-Contrast Problems [Seite 206]
5.12.1 - 1 Summary [Seite 206]
5.12.2 - 2 Introduction [Seite 206]
5.12.3 - 3 Problem Setting and Domain Decomposition Framework [Seite 207]
5.12.4 - 4 Coarse-Space-Completing Eigenvalue Problem and Stability Estimates [Seite 209]
5.12.5 - 5 Numerical Results [Seite 211]
5.12.6 - Bibliography [Seite 213]
5.13 - Weighted Poincaré Inequalities and Applications in Domain Decomposition [Seite 214]
5.13.1 - 1 Introduction [Seite 214]
5.13.2 - 2 Weighted Poincaré Inequalities [Seite 215]
5.13.3 - 3 Explicit Dependence on Geometrical Parameters [Seite 217]
5.13.4 - Bibliography [Seite 220]
5.14 - Technical Tools for Boundary Layers and Applications to Heterogeneous Coefficients [Seite 222]
5.14.1 - 1 Summary [Seite 222]
5.14.2 - 2 Introduction and Assumptions [Seite 222]
5.14.3 - 3 Technical Tools for Layers [Seite 224]
5.14.3.1 - 3.1 Technical Tools for DDMs [Seite 225]
5.14.4 - 4 Dual-Primal Formulation [Seite 226]
5.14.5 - 5 FETI-DP Preconditioner [Seite 228]
5.14.6 - Bibliography [Seite 229]
5.15 - Coarse Spaces over the Ages [Seite 230]
5.15.1 - 1 Introduction [Seite 230]
5.15.2 - 2 Local Nullspace and Bounded Energy Conditions [Seite 230]
5.15.3 - 3 Some Early Domain Decomposition Methods [Seite 232]
5.15.4 - 4 Balancing Domain Decomposition (BDD) and FETI [Seite 233]
5.15.5 - 5 BDDC and FETI-DP [Seite 234]
5.15.6 - 6 Adaptive Methods by Enriching the Coarse Space [Seite 235]
5.15.7 - Bibliography [Seite 235]
5.16 - FETI-DP for Stokes-Mortar-Darcy Systems [Seite 238]
5.16.1 - 1 Introduction and Problem Setting [Seite 238]
5.16.2 - 2 Weak Formulation [Seite 239]
5.16.3 - 3 Discretization and Decomposition [Seite 240]
5.16.4 - 4 Dual Formulation [Seite 242]
5.16.4.1 - 4.1 Dirichlet Preconditioner [Seite 243]
5.16.5 - 5 Numerical Results [Seite 244]
5.16.6 - Bibliography [Seite 245]
5.17 - Multigrid Methods for Elliptic Obstacle Problems on 2D Bisection Grids [Seite 246]
5.17.1 - 1 Introduction [Seite 246]
5.17.2 - 2 Constraint Decomposition Methods [Seite 247]
5.17.3 - 3 A Constraint Decomposition on Bisection Grids [Seite 248]
5.17.4 - 4 Numerical Experiments [Seite 252]
5.17.5 - Bibliography [Seite 252]
5.18 - How Close to the Fully Viscous Solution Can One Get with Inviscid Approximations in Subregions ? [Seite 254]
5.18.1 - 1 Introduction [Seite 254]
5.18.2 - 2 Model Problem [Seite 255]
5.18.3 - 3 Factorization of the Differential Operator [Seite 256]
5.18.4 - 4 Optimal Coupling Conditions and Approximations [Seite 257]
5.18.5 - 5 Numerical Asymptotic Study [Seite 258]
5.18.6 - 6 Conclusions [Seite 260]
5.18.7 - Bibliography [Seite 260]
5.19 - Schwarz Waveform Relaxation Algorithms with Nonlinear Transmission Conditions for Reaction-Diffusion Equations [Seite 262]
5.19.1 - 1 Introduction [Seite 262]
5.19.2 - 2 Problem Description [Seite 263]
5.19.3 - 3 The Schwarz Waveform Relaxation Algorithm [Seite 263]
5.19.3.1 - 3.1 Non-overlapping Algorithms of Order Zero and Two [Seite 264]
5.19.3.2 - 3.2 Well-Posedness and Convergence [Seite 264]
5.19.4 - 4 Discretization [Seite 265]
5.19.4.1 - 4.1 Nonlinear Transmission Conditions [Seite 265]
5.19.4.2 - 4.2 Implementation of the Iterative Algorithm [Seite 266]
5.19.5 - 5 Numerical Results [Seite 267]
5.19.5.1 - 5.1 A Simple Model in Geological CO2 Storage Modeling [Seite 268]
5.19.6 - Bibliography [Seite 269]
5.20 - Recent Advances in Schwarz Waveform Moving Mesh Methods -- A New Moving Subdomain Method [Seite 270]
5.20.1 - 1 Introduction [Seite 270]
5.20.2 - 2 Moving Meshes [Seite 270]
5.20.3 - 3 Domain Decomposition Strategies [Seite 272]
5.20.3.1 - 3.1 SWR in Physical Co-ordinates -- Existing Methods [Seite 273]
5.20.3.2 - 3.2 SWR in Computational Co-ordinates -- A New Approach [Seite 274]
5.20.4 - 4 Numerical Results and Comments [Seite 275]
5.20.5 - Bibliography [Seite 277]
5.21 - Optimized Schwarz Waveform Relaxation Methods: A Large Scale Numerical Study [Seite 278]
5.21.1 - 1 Introduction [Seite 278]
5.21.2 - 2 Optimized Schwarz Waveform Relaxation [Seite 278]
5.21.3 - 3 Theoretical Results [Seite 279]
5.21.4 - 4 Numerical Experiments [Seite 282]
5.21.5 - 5 Conclusions [Seite 284]
5.21.6 - Bibliography [Seite 285]
5.22 - Optimized Schwarz Methods for Maxwell's Equations with Non-zero Electric Conductivity [Seite 286]
5.22.1 - 1 Introduction [Seite 286]
5.22.2 - 2 Schwarz Methods for Maxwell's Equations [Seite 286]
5.22.3 - 3 Analysis for Non-zero Electric Conductivity [Seite 288]
5.22.4 - 4 Numerical Results [Seite 291]
5.22.5 - 5 Conclusion [Seite 293]
5.22.6 - Bibliography [Seite 293]
5.23 - Robust Boundary Element Domain Decomposition Solvers in Acoustics [Seite 294]
5.23.1 - 1 Introduction [Seite 294]
5.23.2 - 2 Formulation of the Domain Decomposition Approach [Seite 294]
5.23.3 - 3 Construction of Preconditioners [Seite 297]
5.23.3.1 - 3.1 Local Preconditioners [Seite 297]
5.23.3.2 - 3.2 Global Preconditioners [Seite 298]
5.23.4 - 4 Numerical Examples [Seite 299]
5.23.4.1 - 4.1 Local Preconditioners [Seite 299]
5.23.4.2 - 4.2 Global Preconditioners [Seite 300]
5.23.5 - Bibliography [Seite 301]
5.24 - A Newton Based Fluid--Structure Interaction Solver with Algebraic Multigrid Methods on Hybrid Meshes [Seite 302]
5.24.1 - 1 Problem Setting of the Fluid--Structure Interaction [Seite 302]
5.24.1.1 - 1.1 Geometrical Description [Seite 302]
5.24.1.2 - 1.2 The Physical Model [Seite 303]
5.24.1.3 - 1.3 Reformulation of the Model [Seite 304]
5.24.1.4 - 1.4 Time Semi-Discretized Weak Formulations [Seite 305]
5.24.1.4.1 - Time Semi-discretized Structure Weak Formulation [Seite 305]
5.24.1.4.2 - Time Semi-discretized Fluid Weak Formulation [Seite 305]
5.24.1.4.3 - The Variational Form of the Interface Equation [Seite 306]
5.24.2 - 2 Newton's Method for the Interface Equation [Seite 307]
5.24.3 - 3 Finite Element Discretization on Hybrid Meshes [Seite 307]
5.24.4 - 4 AMG for the Structure and the Fluid Sub-problems [Seite 307]
5.24.5 - 5 Numerical Results [Seite 308]
5.24.6 - Bibliography [Seite 309]
5.25 - Coupled FE/BE Formulations for the Fluid--Structure Interaction [Seite 310]
5.25.1 - 1 Introduction [Seite 310]
5.25.2 - 2 Integral Equations and Variational Formulations [Seite 311]
5.25.3 - 3 Symmetric Coupling of Finite and Boundary Elements [Seite 312]
5.25.4 - 4 Nonsymmetric Finite and Boundary Element Coupling [Seite 314]
5.25.4.1 - 4.1 A Second Kind Boundary Integral Equation Approach [Seite 314]
5.25.4.2 - 4.2 A First Kind Boundary Integral Equation Approach [Seite 315]
5.25.5 - 5 Conclusions [Seite 316]
5.25.6 - Bibliography [Seite 317]
5.26 - Domain Decomposition Solvers for Frequency-Domain Finite Element Equations [Seite 318]
5.26.1 - 1 Introduction [Seite 318]
5.26.2 - 2 Frequency-Domain Finite Element Equations [Seite 319]
5.26.3 - 3 Domain Decomposition Solver [Seite 321]
5.26.4 - 4 A Symmetric and Indefinite Reformulation [Seite 322]
5.26.5 - 5 Conclusions, Outlook, and Acknowledgments [Seite 324]
5.26.6 - Bibliography [Seite 324]
5.27 - Deriving the X-Z Identity from Auxiliary Space Method [Seite 326]
5.27.1 - 1 Iterative Methods [Seite 326]
5.27.2 - 2 Auxiliary Space Method [Seite 327]
5.27.3 - 3 Auxiliary Spaces of Product Type [Seite 329]
5.27.4 - 4 Method of Subspace Correction [Seite 331]
5.27.5 - Bibliography [Seite 333]
5.28 - A Near-Optimal Hierarchical Estimate Based Adaptive Finite Element Method for Obstacle Problems [Seite 334]
5.28.1 - 1 Introduction [Seite 334]
5.28.2 - 2 A Near-Optimal Hierarchical Error Estimate [Seite 335]
5.28.3 - 3 An Adaptive Finite Element Method [Seite 337]
5.28.4 - 4 Numerical Experiments [Seite 338]
5.28.5 - Bibliography [Seite 340]
5.29 - Efficient Parallel Preconditioners for High-Order Finite Element Discretizations of H(grad) and H(curl) Problems [Seite 342]
5.29.1 - 1 Introduction [Seite 342]
5.29.2 - 2 A Parallel Preconditioner for the H(grad) System [Seite 343]
5.29.2.1 - 2.1 A Parallel AMG Preconditioner [Seite 344]
5.29.2.2 - 2.2 Numerical Experiments [Seite 346]
5.29.3 - 3 A Parallel Preconditioner for the H(curl) Problem [Seite 347]
5.29.3.1 - 3.1 A Parallel Preconditioner for (5) [Seite 347]
5.29.3.2 - 3.2 Numerical Results [Seite 348]
5.29.4 - Bibliography [Seite 349]
6 - Part III Contributed Presentations [Seite 350]
6.1 - A Simple Uniformly Convergent Iterative Method for the Non-symmetric Incomplete Interior Penalty Discontinuous Galerkin Discretization [Seite 351]
6.1.1 - 1 Introduction [Seite 351]
6.1.2 - 2 Interior Penalty Discontinuous Galerkin Methods [Seite 352]
6.1.3 - 3 Space Decomposition [Seite 354]
6.1.3.1 - 3.1 Matrix Representation of the DG Bilinear Forms [Seite 355]
6.1.4 - 4 A Uniformly Convergent Iterative Method [Seite 356]
6.1.5 - 5 Numerical Results [Seite 356]
6.1.6 - Bibliography [Seite 358]
6.2 - A Study of Prolongation OperatorsBetween Non-nested Meshes [Seite 359]
6.2.1 - 1 Introduction [Seite 359]
6.2.2 - 2 Multilevel Preconditioners Based on Non-nested Meshes [Seite 360]
6.2.3 - 3 Looking for Suitable Prolongation Operators [Seite 362]
6.2.4 - 4 Numerical Results [Seite 364]
6.2.5 - Bibliography [Seite 366]
6.3 - A Parallel Schwarz Method for Multiple Scattering Problems [Seite 367]
6.3.1 - 1 Introduction [Seite 367]
6.3.2 - 2 Exterior Helmholtz Problem and Schwarz Method [Seite 368]
6.3.2.1 - 2.1 Domain Decomposition [Seite 368]
6.3.2.2 - 2.2 A Parallel Schwarz Method [Seite 368]
6.3.3 - 3 Multiple DtN Operator [Seite 369]
6.3.4 - 4 How to Solve Problem (2) [Seite 371]
6.3.5 - 5 Proof of Theorem 1 [Seite 371]
6.3.6 - 6 Concluding Remarks [Seite 373]
6.3.7 - Bibliography [Seite 374]
6.4 - Numerical Method for Antenna Radiation Problem by FDTD Method with PML [Seite 375]
6.4.1 - 1 FDTD Method and PML [Seite 375]
6.4.2 - 2 Basic Formulation of Antenna Problem [Seite 377]
6.4.3 - 3 Application to MRI Problem [Seite 379]
6.4.4 - 4 Summary and Future Problems [Seite 380]
6.4.5 - Bibliography [Seite 381]
6.5 - On Domain Decomposition Algorithms for Contact Problems with Tresca Friction [Seite 382]
6.5.1 - 1 Introduction [Seite 382]
6.5.2 - 2 Contact Problems with Tresca Friction [Seite 382]
6.5.3 - 3 Algorithms and the Implementation [Seite 383]
6.5.4 - 4 Numerical Experiments [Seite 386]
6.5.5 - 5 Conclusions and Comments [Seite 388]
6.5.6 - Bibliography [Seite 388]
6.6 - Numerical Solution of Linear Elliptic Problems with Robin Boundary Conditions by a Least-Squares/Fictitious Domain Method [Seite 390]
6.6.1 - 1 Introduction [Seite 390]
6.6.2 - 2 Formulation of the Boundary Value Problem [Seite 390]
6.6.3 - 3 A Least-Squares/Fictitious Domain Method for the Solution of Problem (1), (2) [Seite 391]
6.6.3.1 - 3.1 A Fictitious Domain Formulation of Problem (1), (2) [Seite 391]
6.6.3.2 - 3.2 A Least-Squares Formulation of Problem (7) [Seite 392]
6.6.4 - 4 On the Conjugate Gradient Solution of the Least-Squares Problem (8) [Seite 392]
6.6.5 - 5 On the Finite Element Implementation of the Least-Squares/ Fictitious Domain Methodology [Seite 394]
6.6.5.1 - 5.1 Generalities [Seite 394]
6.6.5.2 - 5.2 Finite Element Approximation of the Least-Squares Problem (8) [Seite 394]
6.6.6 - 6 Numerical Experiments [Seite 395]
6.6.7 - Bibliography [Seite 397]
6.7 - An Uzawa Domain Decomposition Method for Stokes Problem [Seite 398]
6.7.1 - 1 Introduction [Seite 398]
6.7.2 - 2 Model Problem [Seite 398]
6.7.3 - 3 Uzawa Domain Decomposition for Stokes Problem [Seite 399]
6.7.3.1 - 3.1 Lagrangian Formulation and Dual Problem [Seite 400]
6.7.3.2 - 3.2 Sensitivity Analysis [Seite 401]
6.7.3.3 - 3.3 Uzawa Conjugate Gradient Domain Decomposition Algorithm [Seite 402]
6.7.4 - 4 Numerical Experiments [Seite 403]
6.7.5 - 5 Conclusion [Seite 405]
6.7.6 - Bibliography [Seite 405]
6.8 - A Domain Decomposition Method Combining a Boundary Element Method with a Meshless Local Petrov-Galerkin Method [Seite 406]
6.8.1 - 1 Introduction [Seite 406]
6.8.2 - 2 A DDM Combining BEM with the MLPG Method [Seite 407]
6.8.3 - 3 A Dynamic Relaxation Parameter [Seite 410]
6.8.4 - 4 Numerical Examples [Seite 410]
6.8.5 - 5 Conclusions [Seite 412]
6.8.6 - Bibliography [Seite 412]
6.9 - A Domain Decomposition Method Based on Augmented Lagrangian with a Penalty Term in Three Dimensions [Seite 414]
6.9.1 - 1 Introduction [Seite 414]
6.9.2 - 2 Dual Iterative Substructuring with a Penalty Term [Seite 415]
6.9.3 - 3 Estimate of Condition Number [Seite 418]
6.9.4 - 4 Computational Issues [Seite 419]
6.9.5 - Bibliography [Seite 421]
6.10 - Spectral Element Agglomerate Algebraic Multigrid Methods for Elliptic Problems with High-Contrast Coefficients [Seite 422]
6.10.1 - 1 Summary [Seite 422]
6.10.2 - 2 Introduction [Seite 422]
6.10.3 - 3 Notation and Building Tools [Seite 423]
6.10.4 - 4 Multigrid Method [Seite 426]
6.10.5 - 5 Multilevel Additive Preconditioner (BPX) [Seite 426]
6.10.6 - 6 Condition Number Bounds [Seite 427]
6.10.7 - 7 Numerical Experiments [Seite 427]
6.10.8 - Bibliography [Seite 429]
6.11 - A FETI-DP Formation for the Stokes Problem Without Primal Pressure Components [Seite 430]
6.11.1 - 1 Introduction [Seite 430]
6.11.2 - 2 FETI-DP Formulation [Seite 431]
6.11.2.1 - 2.1 Model Problem [Seite 431]
6.11.2.2 - 2.2 FETI-DP Formulation Without Primal Pressure Components [Seite 432]
6.11.3 - 3 Analysis of a Bound of Condition Number [Seite 435]
6.11.3.1 - 3.1 Lower Bound [Seite 435]
6.11.3.2 - 3.2 Upper Bound [Seite 436]
6.11.4 - Bibliography [Seite 437]
6.12 - Schwarz Waveform Relaxation Methods for Systems of Semi-Linear Reaction-Diffusion Equations [Seite 438]
6.12.1 - 1 Introduction [Seite 438]
6.12.2 - 2 Systems of Semi-linear Reaction Diffusion Equations [Seite 439]
6.12.3 - 3 Schwarz Waveform Relaxation Algorithm [Seite 440]
6.12.4 - 4 Numerical Results [Seite 442]
6.12.4.1 - 4.1 Belousov-Zhabotinsky Equations [Seite 442]
6.12.4.2 - 4.2 FitzHugh-Nagumo Equations [Seite 443]
6.12.4.3 - 4.3 Lotka-Volterra Equations [Seite 443]
6.12.5 - 5 Conclusions [Seite 445]
6.12.6 - Bibliography [Seite 445]
6.13 - A Sparse QS-Decomposition for Large Sparse Linear System of Equations [Seite 446]
6.13.1 - 1 Introduction [Seite 446]
6.13.2 - 2 A Quasi-Orthogonal Vector Sequence [Seite 447]
6.13.3 - 3 Layered Group Orthogonalization [Seite 448]
6.13.3.1 - 3.1 Algorithm (LGO) [Seite 448]
6.13.3.2 - 3.2 Matrix representation of LGO [Seite 449]
6.13.4 - 4 LGO Solver and Numerical Experiments [Seite 449]
6.13.5 - 5 A Nested Direct Domain Decomposition Idea [Seite 451]
6.13.6 - Bibliography [Seite 453]
6.14 - Is Additive Schwarz with Harmonic Extension Just Lions' Method in Disguise? [Seite 454]
6.14.1 - 1 The Methods of Lions, AS, RAS and ASH [Seite 454]
6.14.2 - 2 Assumptions and the Main Result [Seite 456]
6.14.3 - 3 Proof of the Main Result [Seite 457]
6.14.4 - 4 Convergence Rate [Seite 459]
6.14.5 - 5 Conclusions [Seite 461]
6.14.6 - Bibliography [Seite 461]
6.15 - Domain Decomposition Methods for a Complementarity Problem [Seite 462]
6.15.1 - 1 Introduction [Seite 462]
6.15.2 - 2 Semismooth Function Approaches for Complementarity Problems [Seite 463]
6.15.2.1 - 2.1 Semismooth Newton Methods [Seite 463]
6.15.2.2 - 2.2 Schwarz Preconditioner [Seite 465]
6.15.3 - 3 Numerical Experiments [Seite 465]
6.15.3.1 - 3.1 One-Level Results [Seite 466]
6.15.3.2 - 3.2 Two-Level Results [Seite 466]
6.15.4 - 4 Some Final Remarks [Seite 467]
6.15.5 - Bibliography [Seite 469]
6.16 - A Posteriori Error Estimates for Semilinear Boundary Control Problems [Seite 470]
6.16.1 - 1 Introduction [Seite 470]
6.16.2 - 2 Finite Elements for Boundary Control Problems [Seite 471]
6.16.3 - 3 A Posteriori Error Estimates [Seite 473]
6.16.4 - Bibliography [Seite 477]
6.17 - Lecture Notes in Computational Science and Engineering [Seite 480]
2 - Contents [Seite 10]
3 - Contributors [Seite 15]
4 - Part I Plenary Presentations [Seite 23]
4.1 - Domain Decomposition and hp-Adaptive Finite Elements [Seite 24]
4.1.1 - 1 Introduction [Seite 24]
4.1.2 - 2 A Posteriori Error Estimate [Seite 25]
4.1.3 - 3 Basis Functions [Seite 26]
4.1.4 - 4 Parallel Adaptive Algorithm [Seite 27]
4.1.5 - 5 DD Solver [Seite 28]
4.1.6 - 6 Numerical Results [Seite 31]
4.1.7 - Bibliography [Seite 34]
4.2 - Domain Decomposition Methods for Electromagnetic Wave Propagation Problems in Heterogeneous Media and Complex Domains [Seite 35]
4.2.1 - 1 Introduction [Seite 35]
4.2.2 - 2 Continuous Problem [Seite 36]
4.2.3 - 3 A Family of Schwarz DD Algorithms [Seite 37]
4.2.4 - 4 Discretization by a High Order DG Method [Seite 38]
4.2.4.1 - 4.1 Discretization of the Monodomain Problem [Seite 38]
4.2.4.2 - 4.2 Discretization of the DD Algorithm [Seite 39]
4.2.4.2.1 - DG Formulation of the Multi-Domain Problem [Seite 39]
4.2.4.2.2 - Formulation of an Interface System [Seite 40]
4.2.5 - 5 Numerical Results [Seite 41]
4.2.5.1 - 5.1 The 2D Case [Seite 41]
4.2.5.2 - 5.2 The 3D Case [Seite 43]
4.2.6 - 6 Ongoing and Future Work [Seite 44]
4.2.7 - Bibliography [Seite 45]
4.3 - N--N Solvers for a DG Discretization for Geometrically Nonconforming Substructures and Discontinuous Coefficients [Seite 47]
4.3.1 - 1 Summary [Seite 47]
4.3.2 - 2 Introduction [Seite 47]
4.3.3 - 3 Differential and Discrete Problems [Seite 49]
4.3.3.1 - 3.1 Differential Problem [Seite 49]
4.3.3.2 - 3.2 Discrete Problem [Seite 49]
4.3.3.3 - 3.3 Schur Complement Problem [Seite 50]
4.3.4 - 4 Notation and the Interface Condition [Seite 53]
4.3.5 - 5 Additive Preconditioners [Seite 55]
4.3.5.1 - 5.1 Local Problems [Seite 55]
4.3.5.2 - 5.2 Coarse Problems [Seite 56]
4.3.5.3 - 5.3 Condition Number Estimate for Tas,I [Seite 56]
4.3.6 - 6 Final Remarks [Seite 57]
4.3.7 - Bibliography [Seite 57]
4.4 - On Adaptive-Multilevel BDDC [Seite 59]
4.4.1 - 1 Introduction [Seite 59]
4.4.2 - 2 Abstract BDDC for a Model Problem [Seite 60]
4.4.2.1 - 2.1 Multilevel BDDC [Seite 61]
4.4.3 - 3 Indicator of the Condition Number Bound [Seite 63]
4.4.4 - 4 Optimal Coarse Degrees of Freedom [Seite 64]
4.4.5 - 5 Adaptive-Multilevel BDDC in 2D [Seite 65]
4.4.6 - 6 Numerical Examples and Conclusion [Seite 66]
4.4.7 - Bibliography [Seite 70]
4.5 - Interpolation Based Local Postprocessing for Adaptive Finite Element Approximations in Electronic Structure Calculations [Seite 71]
4.5.1 - 1 Introduction [Seite 71]
4.5.2 - 2 Interpolation Based Finite Element Postprocessing [Seite 73]
4.5.2.1 - 2.1 Finite Element Discretizations [Seite 74]
4.5.2.2 - 2.2 Interpolation Based Local Postprocessing [Seite 75]
4.5.2.3 - 2.3 Quantum Harmonic Oscillator [Seite 75]
4.5.3 - 3 Applications to Electronic Structure Calculations [Seite 76]
4.5.3.1 - 3.1 Linearization of Kohn--Sham Equation [Seite 76]
4.5.3.2 - 3.2 Experiments [Seite 77]
4.5.3.2.1 - Benzene [Seite 78]
4.5.3.2.2 - Fullerene [Seite 79]
4.5.4 - 4 Concluding Remarks [Seite 80]
4.5.5 - Bibliography [Seite 80]
4.6 - A New a Posteriori Error Estimate for Adaptive Finite Element Methods [Seite 82]
4.6.1 - 1 Introduction [Seite 82]
4.6.2 - 2 A Posteriori Error Estimate [Seite 83]
4.6.3 - 3 Numerical Validation and Applications [Seite 89]
4.6.4 - Bibliography [Seite 92]
4.7 - Space-Time Nonconforming Optimized Schwarz Waveform Relaxation for Heterogeneous Problems and General Geometries [Seite 94]
4.7.1 - 1 Introduction [Seite 94]
4.7.2 - 2 The Continuous OSWR Algorithm [Seite 95]
4.7.3 - 3 Numerical Results [Seite 100]
4.7.4 - 4 Conclusions [Seite 103]
4.7.5 - Bibliography [Seite 105]
4.8 - Convergence Behaviour of Dirichlet--Neumann and Robin Methods for a Nonlinear Transmission Problem [Seite 106]
4.8.1 - 1 Introduction [Seite 106]
4.8.2 - 2 Transmission Problem with Jumping Nonlinearities [Seite 108]
4.8.3 - 3 Nonlinear Dirichlet--Neumann and Robin Methods [Seite 109]
4.8.3.1 - 3.1 The Methods and Their Steklov--Poincaré Formulations [Seite 109]
4.8.3.2 - 3.2 Convergence Results [Seite 110]
4.8.4 - 4 Parameter Studies for the Dirichlet--Neumann Method [Seite 111]
4.8.5 - 5 Parameter Studies for the Robin Method [Seite 114]
4.8.6 - Bibliography [Seite 117]
5 - Part II Minisymposia [Seite 118]
5.1 - Optimal Interface Conditions for an Arbitrary Decomposition into Subdomains [Seite 119]
5.1.1 - 1 Optimal Interface Conditions [Seite 119]
5.1.2 - 2 Notation and Assumptions [Seite 120]
5.1.3 - 3 Construction of the Method [Seite 120]
5.1.4 - 4 Sparsity Pattern [Seite 122]
5.1.5 - 5 Numerical Examples [Seite 124]
5.1.6 - 6 Conclusion [Seite 125]
5.1.7 - Bibliography [Seite 126]
5.2 - Optimized Schwarz Methods for Domains with an Arbitrary Interface [Seite 127]
5.2.1 - 1 Introduction [Seite 127]
5.2.2 - 2 First-Order Boundary Condition [Seite 128]
5.2.3 - 3 Higher-Order Boundary Condition [Seite 130]
5.2.4 - Bibliography [Seite 133]
5.3 - Can the Discretization Modify the Performance of Schwarz Methods? [Seite 135]
5.3.1 - 1 Introduction [Seite 135]
5.3.2 - 2 The Cauchy--Riemann Equations [Seite 135]
5.3.3 - 3 The Positive Definite Helmholtz Equation [Seite 140]
5.3.4 - 4 Conclusions [Seite 141]
5.3.5 - Bibliography [Seite 141]
5.4 - The Pole Condition: A Padé Approximation of the Dirichlet to Neumann Operator [Seite 143]
5.4.1 - 1 Introduction [Seite 143]
5.4.2 - 2 Model Problem [Seite 144]
5.4.3 - 3 The Pole Condition [Seite 145]
5.4.4 - 4 Error Estimate [Seite 147]
5.4.5 - Bibliography [Seite 149]
5.5 - Discontinuous Galerkin and Nonconforming in Time Optimized Schwarz Waveform Relaxation [Seite 151]
5.5.1 - 1 Introduction [Seite 151]
5.5.2 - 2 Local Problem and Time Discontinuous Galerkin [Seite 152]
5.5.3 - 3 The Optimized Schwarz Waveform Relaxation Algorithm Discretized in Time with Different Subdomain Grids [Seite 153]
5.5.4 - 4 Numerical Results [Seite 156]
5.5.5 - 5 Conclusions [Seite 158]
5.5.6 - Bibliography [Seite 158]
5.6 - Two-Level Methods for Blood Flow Simulation [Seite 159]
5.6.1 - 1 Introduction [Seite 159]
5.6.2 - 2 Mathematical Model and Discretization [Seite 159]
5.6.3 - 3 Two-Level Newton and Schwarz Methods [Seite 161]
5.6.4 - 4 Numerical Results [Seite 163]
5.6.5 - 5 Conclusion [Seite 166]
5.6.6 - Bibliography [Seite 166]
5.7 - Newton-Krylov-Schwarz Method for a Spherical Shallow Water Model [Seite 167]
5.7.1 - 1 Introduction [Seite 167]
5.7.2 - 2 Governing Equations [Seite 167]
5.7.3 - 3 Discretizations [Seite 168]
5.7.4 - 4 Nonlinear Solver [Seite 169]
5.7.5 - 5 Numerical Results [Seite 170]
5.7.6 - Bibliography [Seite 172]
5.8 - A Parallel Scalable PETSc-Based Jacobi-Davidson Polynomial Eigensolver with Application in Quantum Dot Simulation [Seite 174]
5.8.1 - 1 Introduction [Seite 174]
5.8.2 - 2 A Description of the ASPJD Algorithm [Seite 175]
5.8.3 - 3 A PETSc-Based ASPJD Polynomial Eigensolver [Seite 177]
5.8.4 - 4 Numerical Results [Seite 178]
5.8.5 - Bibliography [Seite 180]
5.9 - Two-Level Multiplicative Domain Decomposition Algorithm for Recovering the Lamé Coefficient in Biological Tissues [Seite 182]
5.9.1 - 1 Introduction [Seite 182]
5.9.2 - 2 Recovering the Lamé Coefficient in Biological Tissues [Seite 182]
5.9.3 - 3 Lagrange-Newton-Krylov-Schwarz Algorithm [Seite 184]
5.9.4 - 4 Numerical Results and Discussion [Seite 186]
5.9.5 - 5 Concluding Remarks [Seite 188]
5.9.6 - Bibliography [Seite 189]
5.10 - Robust Preconditioner for H(curl) Interface Problems [Seite 190]
5.10.1 - 1 Introduction [Seite 190]
5.10.2 - 2 Regular Decomposition [Seite 191]
5.10.3 - 3 Auxiliary Space Preconditioners [Seite 194]
5.10.4 - 4 Conclusions [Seite 196]
5.10.5 - Bibliography [Seite 196]
5.11 - Mixed Multiscale Finite Element Analysis for Wave Equations Using Global Information [Seite 198]
5.11.1 - 1 Introduction [Seite 198]
5.11.2 - 2 Preliminaries [Seite 199]
5.11.3 - 3 Mixed MsFEM Analysis [Seite 200]
5.11.3.1 - 3.1 Mixed MsFEM Formulation [Seite 200]
5.11.3.2 - 3.2 A Priori Error Estimates for Continuous Time [Seite 202]
5.11.3.3 - 3.3 A Priori Error Estimate for Discrete Time [Seite 204]
5.11.4 - 4 Conclusions [Seite 205]
5.11.5 - Bibliography [Seite 205]
5.12 - A Domain Decomposition Preconditioner for Multiscale High-Contrast Problems [Seite 206]
5.12.1 - 1 Summary [Seite 206]
5.12.2 - 2 Introduction [Seite 206]
5.12.3 - 3 Problem Setting and Domain Decomposition Framework [Seite 207]
5.12.4 - 4 Coarse-Space-Completing Eigenvalue Problem and Stability Estimates [Seite 209]
5.12.5 - 5 Numerical Results [Seite 211]
5.12.6 - Bibliography [Seite 213]
5.13 - Weighted Poincaré Inequalities and Applications in Domain Decomposition [Seite 214]
5.13.1 - 1 Introduction [Seite 214]
5.13.2 - 2 Weighted Poincaré Inequalities [Seite 215]
5.13.3 - 3 Explicit Dependence on Geometrical Parameters [Seite 217]
5.13.4 - Bibliography [Seite 220]
5.14 - Technical Tools for Boundary Layers and Applications to Heterogeneous Coefficients [Seite 222]
5.14.1 - 1 Summary [Seite 222]
5.14.2 - 2 Introduction and Assumptions [Seite 222]
5.14.3 - 3 Technical Tools for Layers [Seite 224]
5.14.3.1 - 3.1 Technical Tools for DDMs [Seite 225]
5.14.4 - 4 Dual-Primal Formulation [Seite 226]
5.14.5 - 5 FETI-DP Preconditioner [Seite 228]
5.14.6 - Bibliography [Seite 229]
5.15 - Coarse Spaces over the Ages [Seite 230]
5.15.1 - 1 Introduction [Seite 230]
5.15.2 - 2 Local Nullspace and Bounded Energy Conditions [Seite 230]
5.15.3 - 3 Some Early Domain Decomposition Methods [Seite 232]
5.15.4 - 4 Balancing Domain Decomposition (BDD) and FETI [Seite 233]
5.15.5 - 5 BDDC and FETI-DP [Seite 234]
5.15.6 - 6 Adaptive Methods by Enriching the Coarse Space [Seite 235]
5.15.7 - Bibliography [Seite 235]
5.16 - FETI-DP for Stokes-Mortar-Darcy Systems [Seite 238]
5.16.1 - 1 Introduction and Problem Setting [Seite 238]
5.16.2 - 2 Weak Formulation [Seite 239]
5.16.3 - 3 Discretization and Decomposition [Seite 240]
5.16.4 - 4 Dual Formulation [Seite 242]
5.16.4.1 - 4.1 Dirichlet Preconditioner [Seite 243]
5.16.5 - 5 Numerical Results [Seite 244]
5.16.6 - Bibliography [Seite 245]
5.17 - Multigrid Methods for Elliptic Obstacle Problems on 2D Bisection Grids [Seite 246]
5.17.1 - 1 Introduction [Seite 246]
5.17.2 - 2 Constraint Decomposition Methods [Seite 247]
5.17.3 - 3 A Constraint Decomposition on Bisection Grids [Seite 248]
5.17.4 - 4 Numerical Experiments [Seite 252]
5.17.5 - Bibliography [Seite 252]
5.18 - How Close to the Fully Viscous Solution Can One Get with Inviscid Approximations in Subregions ? [Seite 254]
5.18.1 - 1 Introduction [Seite 254]
5.18.2 - 2 Model Problem [Seite 255]
5.18.3 - 3 Factorization of the Differential Operator [Seite 256]
5.18.4 - 4 Optimal Coupling Conditions and Approximations [Seite 257]
5.18.5 - 5 Numerical Asymptotic Study [Seite 258]
5.18.6 - 6 Conclusions [Seite 260]
5.18.7 - Bibliography [Seite 260]
5.19 - Schwarz Waveform Relaxation Algorithms with Nonlinear Transmission Conditions for Reaction-Diffusion Equations [Seite 262]
5.19.1 - 1 Introduction [Seite 262]
5.19.2 - 2 Problem Description [Seite 263]
5.19.3 - 3 The Schwarz Waveform Relaxation Algorithm [Seite 263]
5.19.3.1 - 3.1 Non-overlapping Algorithms of Order Zero and Two [Seite 264]
5.19.3.2 - 3.2 Well-Posedness and Convergence [Seite 264]
5.19.4 - 4 Discretization [Seite 265]
5.19.4.1 - 4.1 Nonlinear Transmission Conditions [Seite 265]
5.19.4.2 - 4.2 Implementation of the Iterative Algorithm [Seite 266]
5.19.5 - 5 Numerical Results [Seite 267]
5.19.5.1 - 5.1 A Simple Model in Geological CO2 Storage Modeling [Seite 268]
5.19.6 - Bibliography [Seite 269]
5.20 - Recent Advances in Schwarz Waveform Moving Mesh Methods -- A New Moving Subdomain Method [Seite 270]
5.20.1 - 1 Introduction [Seite 270]
5.20.2 - 2 Moving Meshes [Seite 270]
5.20.3 - 3 Domain Decomposition Strategies [Seite 272]
5.20.3.1 - 3.1 SWR in Physical Co-ordinates -- Existing Methods [Seite 273]
5.20.3.2 - 3.2 SWR in Computational Co-ordinates -- A New Approach [Seite 274]
5.20.4 - 4 Numerical Results and Comments [Seite 275]
5.20.5 - Bibliography [Seite 277]
5.21 - Optimized Schwarz Waveform Relaxation Methods: A Large Scale Numerical Study [Seite 278]
5.21.1 - 1 Introduction [Seite 278]
5.21.2 - 2 Optimized Schwarz Waveform Relaxation [Seite 278]
5.21.3 - 3 Theoretical Results [Seite 279]
5.21.4 - 4 Numerical Experiments [Seite 282]
5.21.5 - 5 Conclusions [Seite 284]
5.21.6 - Bibliography [Seite 285]
5.22 - Optimized Schwarz Methods for Maxwell's Equations with Non-zero Electric Conductivity [Seite 286]
5.22.1 - 1 Introduction [Seite 286]
5.22.2 - 2 Schwarz Methods for Maxwell's Equations [Seite 286]
5.22.3 - 3 Analysis for Non-zero Electric Conductivity [Seite 288]
5.22.4 - 4 Numerical Results [Seite 291]
5.22.5 - 5 Conclusion [Seite 293]
5.22.6 - Bibliography [Seite 293]
5.23 - Robust Boundary Element Domain Decomposition Solvers in Acoustics [Seite 294]
5.23.1 - 1 Introduction [Seite 294]
5.23.2 - 2 Formulation of the Domain Decomposition Approach [Seite 294]
5.23.3 - 3 Construction of Preconditioners [Seite 297]
5.23.3.1 - 3.1 Local Preconditioners [Seite 297]
5.23.3.2 - 3.2 Global Preconditioners [Seite 298]
5.23.4 - 4 Numerical Examples [Seite 299]
5.23.4.1 - 4.1 Local Preconditioners [Seite 299]
5.23.4.2 - 4.2 Global Preconditioners [Seite 300]
5.23.5 - Bibliography [Seite 301]
5.24 - A Newton Based Fluid--Structure Interaction Solver with Algebraic Multigrid Methods on Hybrid Meshes [Seite 302]
5.24.1 - 1 Problem Setting of the Fluid--Structure Interaction [Seite 302]
5.24.1.1 - 1.1 Geometrical Description [Seite 302]
5.24.1.2 - 1.2 The Physical Model [Seite 303]
5.24.1.3 - 1.3 Reformulation of the Model [Seite 304]
5.24.1.4 - 1.4 Time Semi-Discretized Weak Formulations [Seite 305]
5.24.1.4.1 - Time Semi-discretized Structure Weak Formulation [Seite 305]
5.24.1.4.2 - Time Semi-discretized Fluid Weak Formulation [Seite 305]
5.24.1.4.3 - The Variational Form of the Interface Equation [Seite 306]
5.24.2 - 2 Newton's Method for the Interface Equation [Seite 307]
5.24.3 - 3 Finite Element Discretization on Hybrid Meshes [Seite 307]
5.24.4 - 4 AMG for the Structure and the Fluid Sub-problems [Seite 307]
5.24.5 - 5 Numerical Results [Seite 308]
5.24.6 - Bibliography [Seite 309]
5.25 - Coupled FE/BE Formulations for the Fluid--Structure Interaction [Seite 310]
5.25.1 - 1 Introduction [Seite 310]
5.25.2 - 2 Integral Equations and Variational Formulations [Seite 311]
5.25.3 - 3 Symmetric Coupling of Finite and Boundary Elements [Seite 312]
5.25.4 - 4 Nonsymmetric Finite and Boundary Element Coupling [Seite 314]
5.25.4.1 - 4.1 A Second Kind Boundary Integral Equation Approach [Seite 314]
5.25.4.2 - 4.2 A First Kind Boundary Integral Equation Approach [Seite 315]
5.25.5 - 5 Conclusions [Seite 316]
5.25.6 - Bibliography [Seite 317]
5.26 - Domain Decomposition Solvers for Frequency-Domain Finite Element Equations [Seite 318]
5.26.1 - 1 Introduction [Seite 318]
5.26.2 - 2 Frequency-Domain Finite Element Equations [Seite 319]
5.26.3 - 3 Domain Decomposition Solver [Seite 321]
5.26.4 - 4 A Symmetric and Indefinite Reformulation [Seite 322]
5.26.5 - 5 Conclusions, Outlook, and Acknowledgments [Seite 324]
5.26.6 - Bibliography [Seite 324]
5.27 - Deriving the X-Z Identity from Auxiliary Space Method [Seite 326]
5.27.1 - 1 Iterative Methods [Seite 326]
5.27.2 - 2 Auxiliary Space Method [Seite 327]
5.27.3 - 3 Auxiliary Spaces of Product Type [Seite 329]
5.27.4 - 4 Method of Subspace Correction [Seite 331]
5.27.5 - Bibliography [Seite 333]
5.28 - A Near-Optimal Hierarchical Estimate Based Adaptive Finite Element Method for Obstacle Problems [Seite 334]
5.28.1 - 1 Introduction [Seite 334]
5.28.2 - 2 A Near-Optimal Hierarchical Error Estimate [Seite 335]
5.28.3 - 3 An Adaptive Finite Element Method [Seite 337]
5.28.4 - 4 Numerical Experiments [Seite 338]
5.28.5 - Bibliography [Seite 340]
5.29 - Efficient Parallel Preconditioners for High-Order Finite Element Discretizations of H(grad) and H(curl) Problems [Seite 342]
5.29.1 - 1 Introduction [Seite 342]
5.29.2 - 2 A Parallel Preconditioner for the H(grad) System [Seite 343]
5.29.2.1 - 2.1 A Parallel AMG Preconditioner [Seite 344]
5.29.2.2 - 2.2 Numerical Experiments [Seite 346]
5.29.3 - 3 A Parallel Preconditioner for the H(curl) Problem [Seite 347]
5.29.3.1 - 3.1 A Parallel Preconditioner for (5) [Seite 347]
5.29.3.2 - 3.2 Numerical Results [Seite 348]
5.29.4 - Bibliography [Seite 349]
6 - Part III Contributed Presentations [Seite 350]
6.1 - A Simple Uniformly Convergent Iterative Method for the Non-symmetric Incomplete Interior Penalty Discontinuous Galerkin Discretization [Seite 351]
6.1.1 - 1 Introduction [Seite 351]
6.1.2 - 2 Interior Penalty Discontinuous Galerkin Methods [Seite 352]
6.1.3 - 3 Space Decomposition [Seite 354]
6.1.3.1 - 3.1 Matrix Representation of the DG Bilinear Forms [Seite 355]
6.1.4 - 4 A Uniformly Convergent Iterative Method [Seite 356]
6.1.5 - 5 Numerical Results [Seite 356]
6.1.6 - Bibliography [Seite 358]
6.2 - A Study of Prolongation OperatorsBetween Non-nested Meshes [Seite 359]
6.2.1 - 1 Introduction [Seite 359]
6.2.2 - 2 Multilevel Preconditioners Based on Non-nested Meshes [Seite 360]
6.2.3 - 3 Looking for Suitable Prolongation Operators [Seite 362]
6.2.4 - 4 Numerical Results [Seite 364]
6.2.5 - Bibliography [Seite 366]
6.3 - A Parallel Schwarz Method for Multiple Scattering Problems [Seite 367]
6.3.1 - 1 Introduction [Seite 367]
6.3.2 - 2 Exterior Helmholtz Problem and Schwarz Method [Seite 368]
6.3.2.1 - 2.1 Domain Decomposition [Seite 368]
6.3.2.2 - 2.2 A Parallel Schwarz Method [Seite 368]
6.3.3 - 3 Multiple DtN Operator [Seite 369]
6.3.4 - 4 How to Solve Problem (2) [Seite 371]
6.3.5 - 5 Proof of Theorem 1 [Seite 371]
6.3.6 - 6 Concluding Remarks [Seite 373]
6.3.7 - Bibliography [Seite 374]
6.4 - Numerical Method for Antenna Radiation Problem by FDTD Method with PML [Seite 375]
6.4.1 - 1 FDTD Method and PML [Seite 375]
6.4.2 - 2 Basic Formulation of Antenna Problem [Seite 377]
6.4.3 - 3 Application to MRI Problem [Seite 379]
6.4.4 - 4 Summary and Future Problems [Seite 380]
6.4.5 - Bibliography [Seite 381]
6.5 - On Domain Decomposition Algorithms for Contact Problems with Tresca Friction [Seite 382]
6.5.1 - 1 Introduction [Seite 382]
6.5.2 - 2 Contact Problems with Tresca Friction [Seite 382]
6.5.3 - 3 Algorithms and the Implementation [Seite 383]
6.5.4 - 4 Numerical Experiments [Seite 386]
6.5.5 - 5 Conclusions and Comments [Seite 388]
6.5.6 - Bibliography [Seite 388]
6.6 - Numerical Solution of Linear Elliptic Problems with Robin Boundary Conditions by a Least-Squares/Fictitious Domain Method [Seite 390]
6.6.1 - 1 Introduction [Seite 390]
6.6.2 - 2 Formulation of the Boundary Value Problem [Seite 390]
6.6.3 - 3 A Least-Squares/Fictitious Domain Method for the Solution of Problem (1), (2) [Seite 391]
6.6.3.1 - 3.1 A Fictitious Domain Formulation of Problem (1), (2) [Seite 391]
6.6.3.2 - 3.2 A Least-Squares Formulation of Problem (7) [Seite 392]
6.6.4 - 4 On the Conjugate Gradient Solution of the Least-Squares Problem (8) [Seite 392]
6.6.5 - 5 On the Finite Element Implementation of the Least-Squares/ Fictitious Domain Methodology [Seite 394]
6.6.5.1 - 5.1 Generalities [Seite 394]
6.6.5.2 - 5.2 Finite Element Approximation of the Least-Squares Problem (8) [Seite 394]
6.6.6 - 6 Numerical Experiments [Seite 395]
6.6.7 - Bibliography [Seite 397]
6.7 - An Uzawa Domain Decomposition Method for Stokes Problem [Seite 398]
6.7.1 - 1 Introduction [Seite 398]
6.7.2 - 2 Model Problem [Seite 398]
6.7.3 - 3 Uzawa Domain Decomposition for Stokes Problem [Seite 399]
6.7.3.1 - 3.1 Lagrangian Formulation and Dual Problem [Seite 400]
6.7.3.2 - 3.2 Sensitivity Analysis [Seite 401]
6.7.3.3 - 3.3 Uzawa Conjugate Gradient Domain Decomposition Algorithm [Seite 402]
6.7.4 - 4 Numerical Experiments [Seite 403]
6.7.5 - 5 Conclusion [Seite 405]
6.7.6 - Bibliography [Seite 405]
6.8 - A Domain Decomposition Method Combining a Boundary Element Method with a Meshless Local Petrov-Galerkin Method [Seite 406]
6.8.1 - 1 Introduction [Seite 406]
6.8.2 - 2 A DDM Combining BEM with the MLPG Method [Seite 407]
6.8.3 - 3 A Dynamic Relaxation Parameter [Seite 410]
6.8.4 - 4 Numerical Examples [Seite 410]
6.8.5 - 5 Conclusions [Seite 412]
6.8.6 - Bibliography [Seite 412]
6.9 - A Domain Decomposition Method Based on Augmented Lagrangian with a Penalty Term in Three Dimensions [Seite 414]
6.9.1 - 1 Introduction [Seite 414]
6.9.2 - 2 Dual Iterative Substructuring with a Penalty Term [Seite 415]
6.9.3 - 3 Estimate of Condition Number [Seite 418]
6.9.4 - 4 Computational Issues [Seite 419]
6.9.5 - Bibliography [Seite 421]
6.10 - Spectral Element Agglomerate Algebraic Multigrid Methods for Elliptic Problems with High-Contrast Coefficients [Seite 422]
6.10.1 - 1 Summary [Seite 422]
6.10.2 - 2 Introduction [Seite 422]
6.10.3 - 3 Notation and Building Tools [Seite 423]
6.10.4 - 4 Multigrid Method [Seite 426]
6.10.5 - 5 Multilevel Additive Preconditioner (BPX) [Seite 426]
6.10.6 - 6 Condition Number Bounds [Seite 427]
6.10.7 - 7 Numerical Experiments [Seite 427]
6.10.8 - Bibliography [Seite 429]
6.11 - A FETI-DP Formation for the Stokes Problem Without Primal Pressure Components [Seite 430]
6.11.1 - 1 Introduction [Seite 430]
6.11.2 - 2 FETI-DP Formulation [Seite 431]
6.11.2.1 - 2.1 Model Problem [Seite 431]
6.11.2.2 - 2.2 FETI-DP Formulation Without Primal Pressure Components [Seite 432]
6.11.3 - 3 Analysis of a Bound of Condition Number [Seite 435]
6.11.3.1 - 3.1 Lower Bound [Seite 435]
6.11.3.2 - 3.2 Upper Bound [Seite 436]
6.11.4 - Bibliography [Seite 437]
6.12 - Schwarz Waveform Relaxation Methods for Systems of Semi-Linear Reaction-Diffusion Equations [Seite 438]
6.12.1 - 1 Introduction [Seite 438]
6.12.2 - 2 Systems of Semi-linear Reaction Diffusion Equations [Seite 439]
6.12.3 - 3 Schwarz Waveform Relaxation Algorithm [Seite 440]
6.12.4 - 4 Numerical Results [Seite 442]
6.12.4.1 - 4.1 Belousov-Zhabotinsky Equations [Seite 442]
6.12.4.2 - 4.2 FitzHugh-Nagumo Equations [Seite 443]
6.12.4.3 - 4.3 Lotka-Volterra Equations [Seite 443]
6.12.5 - 5 Conclusions [Seite 445]
6.12.6 - Bibliography [Seite 445]
6.13 - A Sparse QS-Decomposition for Large Sparse Linear System of Equations [Seite 446]
6.13.1 - 1 Introduction [Seite 446]
6.13.2 - 2 A Quasi-Orthogonal Vector Sequence [Seite 447]
6.13.3 - 3 Layered Group Orthogonalization [Seite 448]
6.13.3.1 - 3.1 Algorithm (LGO) [Seite 448]
6.13.3.2 - 3.2 Matrix representation of LGO [Seite 449]
6.13.4 - 4 LGO Solver and Numerical Experiments [Seite 449]
6.13.5 - 5 A Nested Direct Domain Decomposition Idea [Seite 451]
6.13.6 - Bibliography [Seite 453]
6.14 - Is Additive Schwarz with Harmonic Extension Just Lions' Method in Disguise? [Seite 454]
6.14.1 - 1 The Methods of Lions, AS, RAS and ASH [Seite 454]
6.14.2 - 2 Assumptions and the Main Result [Seite 456]
6.14.3 - 3 Proof of the Main Result [Seite 457]
6.14.4 - 4 Convergence Rate [Seite 459]
6.14.5 - 5 Conclusions [Seite 461]
6.14.6 - Bibliography [Seite 461]
6.15 - Domain Decomposition Methods for a Complementarity Problem [Seite 462]
6.15.1 - 1 Introduction [Seite 462]
6.15.2 - 2 Semismooth Function Approaches for Complementarity Problems [Seite 463]
6.15.2.1 - 2.1 Semismooth Newton Methods [Seite 463]
6.15.2.2 - 2.2 Schwarz Preconditioner [Seite 465]
6.15.3 - 3 Numerical Experiments [Seite 465]
6.15.3.1 - 3.1 One-Level Results [Seite 466]
6.15.3.2 - 3.2 Two-Level Results [Seite 466]
6.15.4 - 4 Some Final Remarks [Seite 467]
6.15.5 - Bibliography [Seite 469]
6.16 - A Posteriori Error Estimates for Semilinear Boundary Control Problems [Seite 470]
6.16.1 - 1 Introduction [Seite 470]
6.16.2 - 2 Finite Elements for Boundary Control Problems [Seite 471]
6.16.3 - 3 A Posteriori Error Estimates [Seite 473]
6.16.4 - Bibliography [Seite 477]
6.17 - Lecture Notes in Computational Science and Engineering [Seite 480]
