Preface
Contents
Contributors
Part I Plenary Presentations
Domain Decomposition and hp-Adaptive Finite Elements
Randolph E. Bank, Hieu Nguyen
1 Introduction
2 A Posteriori Error Estimate
3 Basis Functions
4 Parallel Adaptive Algorithm
5 DD Solver
6 Numerical Results
Bibliography
Domain Decomposition Methods for Electromagnetic Wave Propagation Problems in Heterogeneous Media and Complex Domains
Victorita Dolean, Mohamed El Bouajaji, Martin J. Gander, Stéphane Lanteri, Ronan Perrussel
1 Introduction
2 Continuous Problem
3 A Family of Schwarz DD Algorithms
4 Discretization by a High Order DG Method
4.1 Discretization of the Monodomain Problem
4.2 Discretization of the DD Algorithm
DG Formulation of the Multi-Domain Problem
Formulation of an Interface System
5 Numerical Results
5.1 The 2D Case
5.2 The 3D Case
6 Ongoing and Future Work
Bibliography
N--N Solvers for a DG Discretization for Geometrically Nonconforming Substructures and Discontinuous Coefficients
Maksymilian Dryja, Juan Galvis, Marcus Sarkis
1 Summary
2 Introduction
3 Differential and Discrete Problems
3.1 Differential Problem
3.2 Discrete Problem
3.3 Schur Complement Problem
4 Notation and the Interface Condition
5 Additive Preconditioners
5.1 Local Problems
5.2 Coarse Problems
5.3 Condition Number Estimate for Tas,I
6 Final Remarks
Bibliography
On Adaptive-Multilevel BDDC
B. Sousedik, J. Mandel
1 Introduction
2 Abstract BDDC for a Model Problem
2.1 Multilevel BDDC
3 Indicator of the Condition Number Bound
4 Optimal Coarse Degrees of Freedom
5 Adaptive-Multilevel BDDC in 2D
6 Numerical Examples and Conclusion
Bibliography
Interpolation Based Local Postprocessing for Adaptive Finite Element Approximations in Electronic Structure Calculations
Jun Fang, Xingyu Gao, Xingao Gong, Aihui Zhou
1 Introduction
2 Interpolation Based Finite Element Postprocessing
2.1 Finite Element Discretizations
2.2 Interpolation Based Local Postprocessing
2.3 Quantum Harmonic Oscillator
3 Applications to Electronic Structure Calculations
3.1 Linearization of Kohn--Sham Equation
3.2 Experiments
Benzene
Fullerene
4 Concluding Remarks
Bibliography
A New a Posteriori Error Estimate for Adaptive Finite Element Methods
Yunqing Huang, Huayi Wei, Wei Yang, Nianyu Yi
1 Introduction
2 A Posteriori Error Estimate
3 Numerical Validation and Applications
Bibliography
Space-Time Nonconforming Optimized Schwarz Waveform Relaxation for Heterogeneous Problems and General Geometries
Laurence Halpern, Caroline Japhet, Jérémie Szeftel
1 Introduction
2 The Continuous OSWR Algorithm
3 Numerical Results
4 Conclusions
Bibliography
Convergence Behaviour of Dirichlet--Neumann and Robin Methods for a Nonlinear Transmission Problem
Heiko Berninger, Ralf Kornhuber, Oliver Sander
1 Introduction
2 Transmission Problem with Jumping Nonlinearities
3 Nonlinear Dirichlet--Neumann and Robin Methods
3.1 The Methods and Their Steklov--Poincaré Formulations
3.2 Convergence Results
4 Parameter Studies for the Dirichlet--Neumann Method
5 Parameter Studies for the Robin Method
Bibliography
Part II Minisymposia
Optimal Interface Conditions for an Arbitrary Decomposition into Subdomains
Martin J. Gander, Felix Kwok
1 Optimal Interface Conditions
2 Notation and Assumptions
3 Construction of the Method
4 Sparsity Pattern
5 Numerical Examples
6 Conclusion
Bibliography
Optimized Schwarz Methods for Domains with an Arbitrary Interface
Shiu Hong Lui
1 Introduction
2 First-Order Boundary Condition
3 Higher-Order Boundary Condition
Bibliography
Can the Discretization Modify the Performance of Schwarz Methods?
Victorita Dolean, Martin J. Gander
1 Introduction
2 The Cauchy--Riemann Equations
3 The Positive Definite Helmholtz Equation
4 Conclusions
Bibliography
The Pole Condition: A Padé Approximation of the Dirichlet to Neumann Operator
Martin J. Gander, Achim Schädle
1 Introduction
2 Model Problem
3 The Pole Condition
4 Error Estimate
Bibliography
Discontinuous Galerkin and Nonconforming in Time Optimized Schwarz Waveform Relaxation
Laurence Halpern, Caroline Japhet, Jérémie Szeftel
1 Introduction
2 Local Problem and Time Discontinuous Galerkin
3 The Optimized Schwarz Waveform Relaxation Algorithm Discretized in Time with Different Subdomain Grids
4 Numerical Results
5 Conclusions
Bibliography
Two-Level Methods for Blood Flow Simulation
Andrew T. Barker, Xiao-Chuan Cai
1 Introduction
2 Mathematical Model and Discretization
3 Two-Level Newton and Schwarz Methods
4 Numerical Results
5 Conclusion
Bibliography
Newton-Krylov-Schwarz Method for a Spherical Shallow Water Model
Chao Yang, Xiao-Chuan Cai
1 Introduction
2 Governing Equations
3 Discretizations
4 Nonlinear Solver
5 Numerical Results
Bibliography
A Parallel Scalable PETSc-Based Jacobi-Davidson Polynomial Eigensolver with Application in Quantum Dot Simulation
Zih-Hao Wei, Feng-Nan Hwang, Tsung-Ming Huang, Weichung Wang
1 Introduction
2 A Description of the ASPJD Algorithm
3 A PETSc-Based ASPJD Polynomial Eigensolver
4 Numerical Results
Bibliography
Two-Level Multiplicative Domain Decomposition Algorithm for Recovering the Lamé Coefficient in Biological Tissues
Si Liu, Xiao-Chuan Cai
1 Introduction
2 Recovering the Lamé Coefficient in Biological Tissues
3 Lagrange-Newton-Krylov-Schwarz Algorithm
4 Numerical Results and Discussion
5 Concluding Remarks
Bibliography
Robust Preconditioner for H(curl) Interface Problems
Jinchao Xu, Yunrong Zhu
1 Introduction
2 Regular Decomposition
3 Auxiliary Space Preconditioners
4 Conclusions
Bibliography
Mixed Multiscale Finite Element Analysis for Wave Equations Using Global Information
Lijian Jiang, Yalchin Efendiev
1 Introduction
2 Preliminaries
3 Mixed MsFEM Analysis
3.1 Mixed MsFEM Formulation
3.2 A Priori Error Estimates for Continuous Time
3.3 A Priori Error Estimate for Discrete Time
4 Conclusions
Bibliography
A Domain Decomposition Preconditioner for Multiscale High-Contrast Problems
Yalchin Efendiev, Juan Galvis
1 Summary
2 Introduction
3 Problem Setting and Domain Decomposition Framework
4 Coarse-Space-Completing Eigenvalue Problem and Stability Estimates
5 Numerical Results
Bibliography
Weighted Poincaré Inequalities and Applications in Domain Decomposition
Clemens Pechstein, Robert Scheichl
1 Introduction
2 Weighted Poincaré Inequalities
3 Explicit Dependence on Geometrical Parameters
Bibliography
Technical Tools for Boundary Layers and Applications to Heterogeneous Coefficients
Maksymilian Dryja, Marcus Sarkis
1 Summary
2 Introduction and Assumptions
3 Technical Tools for Layers
3.1 Technical Tools for DDMs
4 Dual-Primal Formulation
5 FETI-DP Preconditioner
Bibliography
Coarse Spaces over the Ages
Jan Mandel, Bedrich Sousedík
1 Introduction
2 Local Nullspace and Bounded Energy Conditions
3 Some Early Domain Decomposition Methods
4 Balancing Domain Decomposition (BDD) and FETI
5 BDDC and FETI-DP
6 Adaptive Methods by Enriching the Coarse Space
Bibliography
FETI-DP for Stokes-Mortar-Darcy Systems
Juan Galvis, Marcus Sarkis
1 Introduction and Problem Setting
2 Weak Formulation
3 Discretization and Decomposition
4 Dual Formulation
4.1 Dirichlet Preconditioner
5 Numerical Results
Bibliography
Multigrid Methods for Elliptic Obstacle Problems on 2D Bisection Grids
Long Chen, Ricardo H. Nochetto, Chen-Song Zhang
1 Introduction
2 Constraint Decomposition Methods
3 A Constraint Decomposition on Bisection Grids
4 Numerical Experiments
Bibliography
How Close to the Fully Viscous Solution Can One Get with Inviscid Approximations in Subregions ?
Martin J. Gander, Laurence Halpern, Veronique Martin
1 Introduction
2 Model Problem
3 Factorization of the Differential Operator
4 Optimal Coupling Conditions and Approximations
5 Numerical Asymptotic Study
6 Conclusions
Bibliography
Schwarz Waveform Relaxation Algorithms with Nonlinear Transmission Conditions for Reaction-Diffusion Equations
Filipa Caetano, Martin J. Gander, Laurence Halpern, Jérémie Szeftel
1 Introduction
2 Problem Description
3 The Schwarz Waveform Relaxation Algorithm
3.1 Non-overlapping Algorithms of Order Zero and Two
3.2 Well-Posedness and Convergence
4 Discretization
4.1 Nonlinear Transmission Conditions
4.2 Implementation of the Iterative Algorithm
5 Numerical Results
5.1 A Simple Model in Geological CO2 Storage Modeling
Bibliography
Recent Advances in Schwarz Waveform Moving Mesh Methods -- A New Moving Subdomain Method
Ronald D. Haynes
1 Introduction
2 Moving Meshes
3 Domain Decomposition Strategies
3.1 SWR in Physical Co-ordinates -- Existing Methods
3.2 SWR in Computational Co-ordinates -- A New Approach
4 Numerical Results and Comments
Bibliography
Optimized Schwarz Waveform Relaxation Methods: A Large Scale Numerical Study
Martin J. Gander, Loïc Gouarin, Laurence Halpern
1 Introduction
2 Optimized Schwarz Waveform Relaxation
3 Theoretical Results
4 Numerical Experiments
5 Conclusions
Bibliography
Optimized Schwarz Methods for Maxwell's Equations with Non-zero Electric Conductivity
Victorita Dolean, Mohamed El Bouajaji, Martin J. Gander, Stéphane Lanteri
1 Introduction
2 Schwarz Methods for Maxwell's Equations
3 Analysis for Non-zero Electric Conductivity
4 Numerical Results
5 Conclusion
Bibliography
Robust Boundary Element Domain Decomposition Solvers in Acoustics
Olaf Steinbach, Markus Windisch
1 Introduction
2 Formulation of the Domain Decomposition Approach
3 Construction of Preconditioners
3.1 Local Preconditioners
3.2 Global Preconditioners
4 Numerical Examples
4.1 Local Preconditioners
4.2 Global Preconditioners
Bibliography
A Newton Based Fluid--Structure Interaction Solver with Algebraic Multigrid Methods on Hybrid Meshes
Huidong Yang, Walter Zulehner
1 Problem Setting of the Fluid--Structure Interaction
1.1 Geometrical Description
1.2 The Physical Model
1.3 Reformulation of the Model
1.4 Time Semi-Discretized Weak Formulations
Time Semi-discretized Structure Weak Formulation
Time Semi-discretized Fluid Weak Formulation
The Variational Form of the Interface Equation
2 Newton's Method for the Interface Equation
3 Finite Element Discretization on Hybrid Meshes
4 AMG for the Structure and the Fluid Sub-problems
5 Numerical Results
Bibliography
Coupled FE/BE Formulations for the Fluid--Structure Interaction
Günther Of, Olaf Steinbach
1 Introduction
2 Integral Equations and Variational Formulations
3 Symmetric Coupling of Finite and Boundary Elements
4 Nonsymmetric Finite and Boundary Element Coupling
4.1 A Second Kind Boundary Integral Equation Approach
4.2 A First Kind Boundary Integral Equation Approach
5 Conclusions
Bibliography
Domain Decomposition Solvers for Frequency-Domain Finite Element Equations
Dylan Copeland, Michael Kolmbauer, Ulrich Langer
1 Introduction
2 Frequency-Domain Finite Element Equations
3 Domain Decomposition Solver
4 A Symmetric and Indefinite Reformulation
5 Conclusions, Outlook, and Acknowledgments
Bibliography
Deriving the X-Z Identity from Auxiliary Space Method
Long Chen
1 Iterative Methods
2 Auxiliary Space Method
3 Auxiliary Spaces of Product Type
4 Method of Subspace Correction
Bibliography
A Near-Optimal Hierarchical Estimate Based Adaptive Finite Element Method for Obstacle Problems
Qingsong Zou
1 Introduction
2 A Near-Optimal Hierarchical Error Estimate
3 An Adaptive Finite Element Method
4 Numerical Experiments
Bibliography
Efficient Parallel Preconditioners for High-Order Finite Element Discretizations of H(grad) and H(curl) Problems
Junxian Wang, Shi Shu, Liuqiang Zhong
1 Introduction
2 A Parallel Preconditioner for the H(grad) System
2.1 A Parallel AMG Preconditioner
2.2 Numerical Experiments
3 A Parallel Preconditioner for the H(curl) Problem
3.1 A Parallel Preconditioner for (5)
3.2 Numerical Results
Bibliography
Part III Contributed Presentations
A Simple Uniformly Convergent Iterative Method for the Non-symmetric Incomplete Interior Penalty Discontinuous Galerkin Discretization
Blanca Ayuso, Ludmil T. Zikatanov
1 Introduction
2 Interior Penalty Discontinuous Galerkin Methods
3 Space Decomposition
3.1 Matrix Representation of the DG Bilinear Forms
4 A Uniformly Convergent Iterative Method
5 Numerical Results
Bibliography
A Study of Prolongation OperatorsBetween Non-nested Meshes
Thomas Dickopf, Rolf Krause
1 Introduction
2 Multilevel Preconditioners Based on Non-nested Meshes
3 Looking for Suitable Prolongation Operators
4 Numerical Results
Bibliography
A Parallel Schwarz Method for Multiple Scattering Problems
Daisuke Koyama
1 Introduction
2 Exterior Helmholtz Problem and Schwarz Method
2.1 Domain Decomposition
2.2 A Parallel Schwarz Method
3 Multiple DtN Operator
4 How to Solve Problem (2)
5 Proof of Theorem 1
6 Concluding Remarks
Bibliography
Numerical Method for Antenna Radiation Problem by FDTD Method with PML
Takashi Kako, Yoshiharu Ohi
1 FDTD Method and PML
2 Basic Formulation of Antenna Problem
3 Application to MRI Problem
4 Summary and Future Problems
Bibliography
On Domain Decomposition Algorithms for Contact Problems with Tresca Friction
Julien Riton, Taoufik Sassi, Radek Kucera
1 Introduction
2 Contact Problems with Tresca Friction
3 Algorithms and the Implementation
4 Numerical Experiments
5 Conclusions and Comments
Bibliography
Numerical Solution of Linear Elliptic Problems with Robin Boundary Conditions by a Least-Squares/Fictitious Domain Method
JRoland Glowinski, Qiaolin He
1 Introduction
2 Formulation of the Boundary Value Problem
3 A Least-Squares/Fictitious Domain Method for the Solution of Problem (1), (2)
3.1 A Fictitious Domain Formulation of Problem (1), (2)
3.2 A Least-Squares Formulation of Problem (7)
4 On the Conjugate Gradient Solution of the Least-Squares Problem (8)
5 On the Finite Element Implementation of the Least-Squares/ Fictitious Domain Methodology
5.1 Generalities
5.2 Finite Element Approximation of the Least-Squares Problem (8)
6 Numerical Experiments
Bibliography
An Uzawa Domain Decomposition Method for Stokes Problem
Jonas Koko, Taoufik Sassi
1 Introduction
2 Model Problem
3 Uzawa Domain Decomposition for Stokes Problem
3.1 Lagrangian Formulation and Dual Problem
3.2 Sensitivity Analysis
3.3 Uzawa Conjugate Gradient Domain Decomposition Algorithm
4 Numerical Experiments
5 Conclusion
Bibliography
A Domain Decomposition Method Combining a Boundary Element Method with a Meshless Local Petrov-Galerkin Method
Li Maojun, Zhu Jialin
1 Introduction
2 A DDM Combining BEM with the MLPG Method
3 A Dynamic Relaxation Parameter
4 Numerical Examples
5 Conclusions
Bibliography
A Domain Decomposition Method Based on Augmented Lagrangian with a Penalty Term in Three Dimensions
Chang-Ock Lee, Eun-Hee Park
1 Introduction
2 Dual Iterative Substructuring with a Penalty Term
3 Estimate of Condition Number
4 Computational Issues
Bibliography
Spectral Element Agglomerate Algebraic Multigrid Methods for Elliptic Problems with High-Contrast Coefficients
Yalchin Efendiev, Juan Galvis, Panayot S. Vassilevski
1 Summary
2 Introduction
3 Notation and Building Tools
4 Multigrid Method
5 Multilevel Additive Preconditioner (BPX)
6 Condition Number Bounds
7 Numerical Experiments
Bibliography
A FETI-DP Formation for the Stokes Problem Without Primal Pressure Components
Hyea Hyun Kim, Chang-Ock Lee
1 Introduction
2 FETI-DP Formulation
2.1 Model Problem
2.2 FETI-DP Formulation Without Primal Pressure Components
3 Analysis of a Bound of Condition Number
3.1 Lower Bound
3.2 Upper Bound
Bibliography
Schwarz Waveform Relaxation Methods for Systems of Semi-Linear Reaction-Diffusion Equations
Stéphane Descombes, Victorita Dolean, Martin J. Gander
1 Introduction
2 Systems of Semi-linear Reaction Diffusion Equations
3 Schwarz Waveform Relaxation Algorithm
4 Numerical Results
4.1 Belousov-Zhabotinsky Equations
4.2 FitzHugh-Nagumo Equations
4.3 Lotka-Volterra Equations
5 Conclusions
Bibliography
A Sparse QS-Decomposition for Large Sparse Linear System of Equations
Wujian Peng, Biswa N. Datta
1 Introduction
2 A Quasi-Orthogonal Vector Sequence
3 Layered Group Orthogonalization
3.1 Algorithm (LGO)
3.2 Matrix representation of LGO
4 LGO Solver and Numerical Experiments
5 A Nested Direct Domain Decomposition Idea
Bibliography
Is Additive Schwarz with Harmonic Extension Just Lions' Method in Disguise?
Felix Kwok
1 The Methods of Lions, AS, RAS and ASH
2 Assumptions and the Main Result
3 Proof of the Main Result
4 Convergence Rate
5 Conclusions
Bibliography
Domain Decomposition Methods for a Complementarity Problem
Haijian Yang, Xiao-Chuan Cai
1 Introduction
2 Semismooth Function Approaches for Complementarity Problems
2.1 Semismooth Newton Methods
2.2 Schwarz Preconditioner
3 Numerical Experiments
3.1 One-Level Results
3.2 Two-Level Results
4 Some Final Remarks
Bibliography
A Posteriori Error Estimates for Semilinear Boundary Control Problems
Yanping Chen, Zuliang Lu
1 Introduction
2 Finite Elements for Boundary Control Problems
3 A Posteriori Error Estimates
Bibliography