Domain Decomposition Methods in Science and Engineering XXVI

✍ Scribed by Susanne C. Brenner, Eric Chung, Axel Klawonn, Felix Kwok, Jinchao Xu, Jun Zou

• Publisher: Springer
• Year: 2023
• Tongue: English
• Leaves: 777
• Series: Lecture Notes in Computational Science and Engineering, 145
• Category: Library

✦ Synopsis

These are the proceedings of the 26th International Conference on Domain Decomposition Methods in Science and Engineering, which was hosted by the Chinese University of Hong Kong and held online in December 2020.

Domain decomposition methods are iterative methods for solving the often very large systems of equations that arise when engineering problems are discretized, frequently using finite elements or other modern techniques. These methods are specifically designed to make effective use of massively parallel, high-performance computing systems.

The book presents both theoretical and computational advances in this domain, reflecting the state of art in 2020.

✦ Table of Contents

Preface
Background of the Conference Series
About the 26th Conference
Acknowledgments
Contents
List of Contributors
Part I Plenary Talks (PT)
Multiscale Model Reduction for a Class of Optimal Control Problems with Highly Oscillating Coefficients
1 Introduction
2 Method description
2.1 Multiscale basis functions
2.2 The proposed method
2.3 Outline of error analysis
3 Numerical results
References
SeveralWays to Achieve Robustness When SolvingWave Propagation Problems
1 Motivation and challenges
1.1 Why the time-harmonic problem in mid and high frequency is hard
2 What is the best coarse space for Helmholtz?
2.1 Spectral coarse spaces for Helmholtz
2.2 Comparison of coarse spaces
3 Can we improve on the auxiliary subspace preconditioner?
4 General conclusions
References
Scalable Hybrid TFETI-DP Methods for Large Boundary Variational Inequalities
1 Introduction
2 Model problem
3 Domain decomposition and discretization
4 TFETI problem
5 Connecting subdomains into clusters
6 Bounds on the spectrum of H and optimality
7 Numerical experiments
7.1 Comparing estimate and experiments
7.2 Comparing linear unpreconditioned H-TFETI-DP and TFETI
7.3 Model variational inequality and elastic body on rigid obstacle
References
Fundamental Coarse Space Components for Schwarz Methods with Crosspoints
1 Introduction
2 Modal analysis of the Schwarz iteration map
3 The special case of 2 × 2 subdomains
4 A new coarse space for parallel Schwarz methods
5 Numerical experiments
6 A note on the optimized Robin parameter
7 Conclusion
References
Nonoverlapping Domain Decomposition Methods for Time HarmonicWave Problems
1 Elliptic equations versus the Helmholtz equation
2 Impedance-based transmission conditions and related DDM
3 Convergence analysis
4 Construction of appropriate impedance operators
5 The problem of cross points
References
Quantitative Analysis of Nonlinear Multifidelity Optimization for Inverse Electrophysiology
1 Introduction
2 Electrophysiological models
3 Inverse problem of Conductivity Identification
4 Multilevel Quasi-Newton Trust-region Method
Quasi-Newton trust-region method
Recursive multilevel trust-region method
5 Numerical results
5.1 Patient-specific geometry
5.2 Convergence study
6 Conclusion
References
Adaptive Space-Time Finite Element and Isogeometric Analysis
1 Introduction
2 Space-time variational formulations
3 Space-time isogeometric analysis
4 Space-time finite element analysis
5 Space-time optimal control
References
Nonoverlapping Domain Decomposition Methods for Saddle Point Problems
1 Introduction
2 Problem setting
3 Domain decomposition
4 Positive definite formulations
4.1 The primal system (6)
4.2 The dual system (12)
5 Condition number estimates
6 Connections and differences
References
Local Residual Minimization Smoothing for Improving Convergence Behavior of a Space-Time Domain Decomposition Method
1 Introduction
2 Flow model problem
3 Local residual minimization
4 Numerical results
5 Conclusions
References
Part II Talks in Minisymposia
GenEO Coarse Spaces for Heterogeneous Indefinite Elliptic Problems
1 Introduction and motivations
2 Discretisation and domain decomposition solver
3 Theoretical results
4 Numerical results
5 Conclusions
References
Inexact Subdomain Solves Using Deflated GMRES for Helmholtz Problems
1 Introduction
2 Model problem, discretisation and preconditioning strategies
3 Numerical results
3.1 Direct subdomain solves
3.2 Inexact subdomain solves
4 Conclusions
References
Non-Overlapping Domain Decomposition Methods with Cross-Points and Padé Approximants for the Helmholtz Equation
1 Introduction
2 Model problem
3 Non-overlapping domain decomposition algorithm
4 Nodal FEM-NDDM and the cross-points problem
5 Numerical results
References
OSDS: A Sweeping Preconditioner for the Helmholtz Equation
1 Statement of the problem and some algorithms
2 Overlapping splitting double sweep
2.1 Substructuring
2.2 Overlapping Splitting Double Sweep preconditioner (OSDS)
3 Numerical results
3.1 Wedge test
3.2 Influence of the overlap
4 Conclusion
References
Decomposition and Preconditioning of Deep Convolutional Neural Networks for Training Acceleration
1 Introduction
2 Proposed approaches
2.1 Decomposing a DCNN into sub-networks
2.2 Preconditioning of DCNNs
3 Experiments
4 Conclusion
References
Numerical Calculation of the Portal Pressure Gradient of the Human Liver With a Domain Decomposition Method
1 Introduction
2 Numerical method
3 Numerical experiments
References
A Parallel Adaptive Finite Element Method for Modeling a Deformable Droplet Travelling in Air
1 Introduction
2 A mathematical model based on the Cahn–Hilliard–Navier–Stokes equations
3 A parallel, semi-implicit solution algorithm based on an adaptive finite element discretization, and an overlapping Schwarz pr
4 Numerical experiments
5 Conclusions
References
On the Effect of Boundary Conditions on the Scalability of Schwarz Methods
1 Introduction
2 Laplace eigenpairs for mixed external conditions
3 Convergence and scalability
4 Numerical experiments
References
On the Asymptotic Optimality of Spectral Coarse Spaces
1 Introduction
2 A perturbation approach
3 Optimizing the coarse-space functions
References
Discrete Analysis of SchwarzWaveform Relaxation for a Simplified Air-Sea Coupling Problem with Nonlinear Transmission Conditions
1 Introduction
2 Model problem for ocean-atmosphere coupling
3 Discretized coupled problem
3.1 Implementation of the surface layer
3.2 SchwarzWaveform Relaxation
4 Convergence analysis
5 Numerical experiments
6 Conclusion
References
A Posteriori Error Estimates in Maximum Norm for Interior Penalty Discontinuous Galerkin Approximation of the Obstacle Problem
1 Introduction
2 The elliptic obstacle problem
3 The Symmetric Interior Penalty method
4 Reliable a posteriori error estimates in the maximum norm
5 Numerical results
5.1 Numerical experiments
References
Spectral Equivalence Properties of Higher-Order Tensor Product Finite Elements
1 Introduction
2 Shape functions
2.1 Edge shape functions
2.2 Face shape functions
2.3 Interior shape functions
2.4 Lowest-order shape functions
3 Spectral equivalence results
3.1 Mass matrix equivalence
3.2 Stiffness matrix equivalence
4 Numerical results
References
Optimizing Transmission Conditions for Multiple Subdomains in the Magnetotelluric Approximation of Maxwell’s Equations
1 Optimized Schwarz for the Magnetotelluric Approximation
2 Optimization for 2, 3, 4, 5 and 6 subdomains
3 Optimization for many subdomains
References
Non-overlapping Spectral Additive Schwarz Methods for HDG and Multiscale Discretizations
1 Introduction
2 Domain decomposition setting
3 NOSAS with exact and inexact solvers
4 Multiscale discretization methods
5 Numerical experiments
References
Robust BPX Solver for Cahn–Hilliard Equations
1 Introduction
2 Numerical approximation
3 Newton’s method
4 Preconditioned iterative solver
5 Conclusions
References
Natural Factor Based Solvers
1 Summary
2 Introduction
3 Crouzeix–Raviart nonconforming finite elements
4 Conforming finite elements
5 The null space method
6 Range null-space hybrid
7 An auxiliary problem and 2 × 2 systems
8 PCG for the block system and numerical experiments
References
A Simple Finite Difference Discretization for Ventcell Transmission Conditions at Cross Points
1 Introduction
2 Optimized Schwarz with Ventcell transmission conditions
3 Numerical experiments
4 Conclusions
References
Cycles in Newton–Raphson Preconditioned by Schwarz (ASPIN and Its Cousins)
1 Introduction
2 Convergence of generic fixed point iterations and Newton–Raphson
3 The fixed point iteration of alternating Schwarz
4 Accelerated alternating Schwarz with guaranteed convergence
References
Should Multilevel Methods for Discontinuous Galerkin Discretizations Use Discontinuous Interpolation Operators?
1 Discontinuous interpolation for a model problem
2 Study of optimal parameters by Local Fourier Analysis
3 Numerical results
4 Conclusion
References
Domain Decomposition in Shallow Water Modelling of Dutch Lakes for Multiple Applications
1 Introduction
1.1 Area of interest
1.2 Previous approaches
1.3 New approach
2 Computational grids
3 Domain decomposition for parallel computing in Delft3D FM
4 Conclusions and outlook
References
A Variational Interpretation of Restricted Additive SchwarzWith Impedance Transmission Condition for the Helmholtz Problem
1 The Helmholtz problem
2 Parallel iterative Schwarz method
3 Variational formulation of RAS with impedance transmission condition (ORAS)
4 Connecting the parallel iterative method with ORAS
5 Numerical results
References
Application of Multilevel BDDC to the Problem of Pressure in Simulations of Incompressible Flow
1 Introduction
2 The pressure-correction method
3 Numerical results
4 Conclusions
References
Predicting the Geometric Location of Critical Edges in Adaptive GDSW Overlapping Domain Decomposition Methods Using Deep Learnin
1 Introduction
2 Model problem and adaptive GDSW
3 Machine learning for adaptive GDSW
4 Numerical results
References
Optimized Coupling Conditions for Discrete Fracture Matrix Models
1 Introduction
2 Model problem
3 Generalized coupling conditions and their optimization
4 Numerical results
5 Conclusion
References
Efficient Monolithic Solvers for Fluid-Structure Interaction Applied to Flapping Membranes
1 Introduction
2 FSI model
3 Numerical solution and physics-based preconditioners
3.1 Newton linearization
3.2 Block structure of linear systems
3.3 Physics-based preconditioner
4 Flapping membranes with elastic solid walls
5 Conclusions and outlook
References
Adaptive Nonlinear Elimination in Nonlinear FETI-DP Methods
1 Introduction
2 Nonlinear FETI-DP
3 A problem-dependent choice of the elimination set
4 Numerical Results
References
Globalization of Nonlinear FETI–DP Methods
1 Introduction
2 Nonlinear FETI-DP
3 Exact differentiable penalty method with nonlinear elimination
4 Numerical results
References
A Multilevel Active-Set Trust-Region (MASTR) Method for Bound Constrained Minimization
1 Introduction
2 The recursive multilevel trust-region (RMTR) method
3 The multilevel active-set trust-region (MASTR) method
Construction of truncated FE spaces
Construction of level-dependent objective functions and feasible sets
4 Numerical results
References
A Multigrid Preconditioner for Jacobian-free Newton–Krylov Methods
1 Introduction
2 Matrix-free multigrid preconditioner
3 Numerical experiments
References
Overlapping DDFV Schwarz Algorithms on Non-Matching Grids
1 Introduction
2 Overlapping DDFV Schwarz algorithm
3 Convergence of overlapping DDFV Schwarz
References
On the Nonlinear Dirichlet–Neumann Method and Preconditioner for Newton’s Method
1 Introduction
2 Nilpotent property and quadratic convergence
3 Mesh independent convergence
4 Dirichlet–Neumann Preconditioned Exact Newton (DNPEN)
5 Conclusion
References
A Nonlinear Optimized Schwarz Preconditioner for Elliptic Optimal Control Problems
1 Introduction
2 An optimized Schwarz method and preconditioner
3 Numerical experiments
4 Further discussion and conclusion
References
SParse Approximate Inverse (SPAI) Based Transmission Conditions for Optimized Algebraic Schwarz Methods
1 Introduction
2 Algebraic Optimized Schwarz methods
3 Sparse approximations of optimal transmission blocks
4 Numerical experiments
Concluding remarks
References
A Parareal Architecture for Very Deep Convolutional Neural Networks
1 Introduction
2 The parareal algorithm
3 Parareal neural networks
4 Application to ResNet-1001
5 Numerical results
References
Construction of 4D Simplex Space-Time Meshes for Local Bisection Schemes
1 Introduction
2 Four-dimensional space-time mesh construction
3 Conforming bisection of space-time simplicial elements
4 Conclusions
References
Coefficient-Robust A Posteriori Error Estimation for H(curl)-elliptic Problems
1 Introduction
1.1 H(curl)-elliptic problems
2 Nodal auxiliary space preconditioning
3 A posteriori error estimates
4 Numerical demonstration of robustness
References
Convergence of Parareal for a Vibrating String with Viscoelastic Damping
1 Model equation for a vibrating string
2 The Parareal algorithm
3 Space mesh requirement for fixed error tolerance
4 Numerical experiment with Parareal
References
Consistent and Asymptotic-Preserving Finite-Volume Robin Transmission Conditions for Singularly Perturbed Elliptic Equations
1 Introduction
2 The continuous problem and non-overlapping DDM
3 Cell-centered finite volume discretization
4 Numerical examples
5 Conclusion
References
Adaptive Schwarz Method for Crouzeix–Raviart Multiscale Problems in 2D
1 Introduction
2 Discrete problem
3 Additive Schwarz Method
Coarse space
Average Schwarz operator T
4 Numerical tests
References
An OverlappingWaveform Relaxation Preconditioner for Economic Optimal Control Problems With State Constraints
1 Introduction
2 The waveform-relation and the preconditioned generalized Newton methods
3 Numerical experiments
References
Optimized Schwarz MethodsWith Data-Sparse Transmission Conditions
1 Introduction and model problem
2 Parallel optimized Schwarz method
3 Data-sparse transmission conditions
References
Space-Time Finite Element Tearing and Interconnecting Domain Decomposition Methods
1 Introduction
2 Space-time finite element methods
3 Tearing and interconnecting domain decomposition methods
4 Numerical results
5 Conclusions
References
Localized Reduced Basis Additive Schwarz Methods
1 Introduction
2 A localized reduced basis additive Schwarz method
2.1 LRBAS as an additive-Schwarz multi-preconditioned CG method
3 Numerical experiment
References
Micromechanics Simulations Coupling the deal.II Software LibraryWith a Parallel FETI-DP Solver
1 Introduction
2 Micromechanical model problem
3 Parallel mesh handling in deal.II
4 Subdomain Neumann matrices in deal.II
5 Numerical results
References
A Three-Level Extension for Fast and Robust Overlapping Schwarz (FROSch) Preconditioners with Reduced Dimensional Coarse Space
1 Fast and Robust Overlapping Schwarz preconditioners
2 Three-level extension
3 Implementation
4 Weak parallel scalability results for the three-Level extension
References
Space-Time Hexahedral Finite Element Methods for Parabolic Evolution Problems
1 Introduction
2 Space-time finite element methods
3 Anisotropic a priori and a posteriori error estimates
4 Numerical results
References
Towards a IETI-DP Solver on Non-Matching Multi-Patch Domains
1 Introduction
2 The problem setting
3 The dG IETI-DP solver
4 Numerical results
References
The Parallel Full Approximation Scheme in Space and Time for a Parabolic Finite Element Problem
1 Introduction
2 PFASST and finite elements in space
2.1 Finite element discretization in space
2.2 The collocation problem and SDC
2.3 The composite collocation problem and PFASST
3 Numerical results
References
A New Coarse Space for a Space-Time Schwarz Waveform Relaxation Method
1 Introduction and model problem
2 New two level Schwarz waveform relaxation
3 Numerical experiments
4 Conclusions
References
On Space-Time Finite Element Domain Decomposition Methods for the Heat Equation
1 Introduction
2 Space-time finite element methods
3 A space-time domain decomposition method
4 Numerical results
5 Conclusions
References
IETI-DP for Conforming Multi-Patch Isogeometric Analysis in Three Dimensions
1 Introduction
2 Model problem and its solution
3 Numerical results
References
Coupling of Navier–Stokes Equations and Their Hydrostatic Versions and Simulation of Riverbend Flow
1 Introduction
2 Governing equations
3 Computational methods
4 Numerical simulation
5 Concluding remarks
References
On the Links Between Observed and Theoretical Convergence Rates for SchwarzWaveform Relaxation Algorithm for the Time-Dependent
1 Context
2 Convergence for problems on an infinite time window
3 Convergence for problems on a finite time window
4 Numerical illustration
5 Conclusion
References
Construction of Grid Operators for Multilevel Solvers: a Neural Network Approach
1 Introduction
2 Problem definition
2.1 Neural networks
2.2 Training transfer operators
3 Training set
3.1 Records
3.2 One-dimensional model
3.3 Two-dimensional model
4 Model training
4.1 Regularization
4.2 Model details
5 Numerical results
6 Conclusion
References
Coarse Corrections for Schwarz methods for Symmetric and Non-symmetric Problems
1 Introduction
2 Two-level RAS with classical and new coarse spaces
3 Numerical results
4 Conclusions
References
A Numerical Algorithm Based on Probing to Find Optimized Transmission Conditions
1 Motivation
2 An algorithm based on probing
3 Numerical experiments
References
Additive Schwarz Preconditioners for C0 Interior Penalty Methods for a State Constrained Elliptic Distributed Optimal Control Problem
1 Introduction
2 The C0 interior penalty method
3 The primal-dual active set algorithm
4 A one-level additive Schwarz preconditioner
5 A two-level additive Schwarz preconditioner
6 Numerical results
7 Conclusion
References
Space-Time Finite Element Methods for Initial Temperature Reconstruction
1 Introduction
2 The related optimal control problem
3 Space-time finite element methods
4 Numerical results
5 Conclusions
References
Numerical Results for an Unconditionally Stable Space-Time Finite Element Method for the Wave Equation
1 Introduction
2 A modified Hilbert transformation
3 Space-time variational formulations
4 Numerical results
5 Conclusions
References
Décomposition de Domaine et Problème de Helmholtz: Thirty Years After and Still Unique
1 Introduction
2 What can we change from Després’ original method?
References
Part III Contributed Talks
Space-Time Parallel Methods for Evolutionary Reaction-Diffusion Problems
1 Introduction and problem setting
2 Domain decomposition splitting methods
3 The parareal method
4 Numerical experiments
References
Parallel Domain Decomposition Solvers for the Time Harmonic Maxwell Equations
1 Introduction
2 Equations and finite element discretization
3 Numerical solution with domain decomposition and preconditioners
3.1 Domain decomposition
3.2 Preconditioner
4 Numerical tests
4.1 Example 1: Block benchmark
4.2 Example 2: Y beam splitter
5 Conclusion
References
Adaptive Finite Element Thin-Plate SplineWith Different Data Distributions
1 Introduction
2 Adaptive refinement
3 Data distribution
4 Numerical experiment
4.1 Results
5 Conclusion
References
A Multirate Accelerated SchwarzWaveform Relaxation Method
1 Introduction
2 Background material
3 A multirate approach
4 An accelerated SWR approach
5 A comparison
6 Conclusions
References
A Convergence Analysis of the Parallel Schwarz Solution of the Continuous Closest Point Method
1 Introduction
2 The closest point method
3 The PS-CPM convergence analysis
4 Numerical simulation
5 Conclusion
References
Dual-Primal Preconditioners for Newton–Krylov Solvers for the Cardiac Bidomain Model
1 Introduction
2 The Bidomain cardiac electrical model
3 Dual-primal Newton–Krylov methods
4 Numerical experiments
5 Conclusion
References
Domain Decomposition Algorithms for Physics-Informed Neural Networks
1 Introduction
2 Physics informed neural networks (PINN)
3 A two-level overlapping algorithm for PINN
4 Numerical results
References
Numerical Study of an Additive Schwarz Preconditioner for a Thin Membrane Diffusion Problem
1 Introduction
2 Additive Schwarz preconditioner
3 Numerical experiments
4 Conclusions
References
Additive Schwarz Methods for Convex Optimization – Convergence Theory and Acceleration
1 Introduction
2 Additive Schwarz methods
3 Convergence theory
4 Acceleration
References
Non-local Impedance Operator for Non-overlapping DDM for the Helmholtz Equation
1 General approach for a two-domain decomposition
2 Quantitative analysis for the wave-guide
3 Finite element computations in a circular geometry
References
Asynchronous Multi-Subdomain MethodsWith Overlap for a Class of Parabolic Problems
1 Introduction
2 Notation and assumptions
2.1 Technical result
2.2 Problem statement
3 Fixed point mappings T
3.1 The linear mapping
3.2 The linear mapping
3.3 Contraction property of T
4 Constants of contraction comparison with respect to the weighted exponential norm
References
Toward a New Fully Algebraic Preconditioner for Symmetric Positive Definite Problems
1 Introduction
2 Definition of the new preconditioner and theory
2.1 Algebraic domain decomposition
2.2 Definition of A+ and related operators
2.3 Two-level preconditioner for A+ with a GenEO coarse space
2.4 New preconditioner for A
3 Numerical illustration
4 Conclusion
References
Aitken–Schwarz Heterogeneous Domain Decomposition for EMT-TS Simulation
1 Introduction
2 EMT and TS modeling
3 EMT and TS Schwarz homogeneous DDM
4 Heterogeneous DDM EMT-TS
5 Conclusion
References
Parareal SchwarzWaveform Relaxation Method for the Time-Periodic Parabolic Problem
1 Introduction and model problem
2 PSWR for time-periodic parabolic problem
3 Numerical experiments
4 Conclusions
References
Acceleration of the Convergence of the Asynchronous RAS Method
1 Introduction
2 Aitken–Schwarz method principles
3 Modeling the asynchronous RAS
4 Regularization of the Aitken acceleration of convergence of the asynchronous RAS
5 Conclusion
References