Domain Decomposition Methods in Science and Engineering XXV 1st edition by Ronald Haynes, Scott MacLachlan, Xiao Chuan Cai, Laurence Halpern, Hyea Hyun Kim, Axel Klawonn, Olof Widlund ISBN 3030567508 9783030567507

Part I: Plenary Talks (PT)

1. Does the Partition of Unity Influence the Convergence of Schwarz Methods?

   • Which Schwarz Methods Need a Partition of Unity?
   • Partitions of Unity for RAS, RASH, ORAS, and ORASH
   • Influence of the Partition of Unity on RAS and RASH in 1D
   • RAS, RASH, ORAS, and ORASH in 2D
   • References

2. Adaptive BDDC Based on Local Eigenproblems

   • Introduction
   • BDDC Basics
   • Localization, Eigenproblems, and Adaptivity
     • Glob-based approach
     • Pair-based approach
     • Three different approaches
   • Simplification of the generalized eigenproblems
   • Optimality of the deluxe scaling
   • Appendix
   • References

3. From Domain Decomposition to Homogenization Theory

   • Introduction
   • Model problem and classical homogenization
   • Novel characterization of the effective coefficient
   • Numerical effective coefficient by domain decomposition
   • Beyond periodicity and scale separation
   • References

4. Robust Model Reduction Discretizations Based on Adaptive BDDC Techniques

   • Introduction
   • Discrete Substructuring Formulation
   • Main Goal of the Paper
   • Model Reduction via BDDC
     • Decaying for the High-Contrast Case
   • Spectral Multiscale Problems inside Substructures
   • Numerical Experiments
   • References

5. Analysis of Double Sweep Optimized Schwarz Methods: the Positive Definite Case

   • Introduction
   • Iteration Matrix of DOSM
   • Eigenvalues of the iteration matrix
   • Roots of the Polynomial Equation
   • Numerical Study of the Convergence Factor
   • References

Part II: Talks in Minisymposia (MT)

1. Dirichlet-Neumann Preconditioning for Stabilized Unfitted Discretization of High Contrast Problems

   • Introduction
   • Basic Notation and Unfitted Stabilized Discretization
   • Dirichlet-Neumann Preconditioner
   • Algebraic Formulation of the DN Preconditioner
   • Numerical Results
   • References

2. Virtual Coarse Spaces for Irregular Subdomain Decompositions

   • Introduction
   • Description of the Preconditioner
     • Virtual coarse space
     • Local spaces and preconditioner
   • Some Numerical Results
   • Conclusions
   • References

3. A Local Coarse Space Correction Leading to a Well-Posed Continuous Neumann-Neumann Method in the Preconditioner

   • Introduction
   • Existence of Iterates for a 2 x 1 Decomposition
   • Existence of Iterates for a 2 x 2 Decomposition
   • References

4. Happy 25th Anniversary DDM! … But How Fast Can the Schwarz Method Solve Your Logo?

   • The DDM Logo Problem and the Schwarz Method
   • Convergence Analysis
   • Numerical Experiments
   • References

5. Additive Schwarz Preconditioners for a State Constrained Elliptic Distributed Optimal Control Problem

   • Introduction
   • The Discrete Problem
   • The Preconditioners
   • A Numerical Example
   • Conclusion
   • References

6. A Parallel Solver for a Preconditioned Space-Time Boundary Element Method for the Heat Equation

   • Introduction
   • Operator Preconditioning
   • Distributed Memory Parallelization
   • Numerical Experiments
   • Conclusion
   • References

7. On Inexact Solvers for the Coarse Problem of BDDC

   • Introduction
   • Motivation
   • Overview of BDDC and Our Inexact Approach
   • Main Results
   • Numerical Results
   • References

8. Simultaneous Approximation Terms for Elastic Wave Equations on Nonuniform Grids

   • Introduction
   • Problem Description
   • Methodology
   • Numerical Examples
   • Summary
   • References

9. Asynchronous One-Level and Two-Level Domain Decomposition Solvers

   • Introduction
   • Domain Decomposition Methods
   • Numerical Experiments
   • References

10. Comparison of Continuous and Discrete Techniques to Apply Coarse Corrections

    • Introduction
    • Discrete Coarse Correction Based on the Residual
    • Continuous Coarse Correction Using a Transmission Problem
    • Comparison of the Discrete and Continuous Techniques
    • Conclusions
    • References

11. On the Scalability of the Parallel Schwarz Method in One-Dimension

    • Introduction and Main Results
    • Proof of Theorem 1 (a)
    • Proof of Theorem 1 (b)
    • References

12. Fully Discrete Schwarz-Waveform Relaxation on Two Bounded Overlapping Subdomains

    • Introduction
    • Semi-Discretized SWR
    • Convergence Analysis
    • Conclusions
    • References

13. Local Spectra of Adaptive Domain Decomposition Methods

    • Introduction
    • Various Adaptive Coarse Spaces in Domain Decomposition
    • Numerical Results
    • References

14. FROSch: A Fast And Robust Overlapping Schwarz Domain Decomposition Preconditioner Based on Xpetra in Trilinos

    • Introduction
    • The GDSW Preconditioner
    • Software Design of the FROSch Library
    • Performance of the New FROSch Software
    • References

15. A Three-Level Extension of the GDSW Overlapping Schwarz Preconditioner in Three Dimensions

    • The Standard GDSW Preconditioner
    • The Three-Level GDSW Preconditioner
    • Implementation and Software Libraries
      • Weak Parallel Scalability of the Three-Level GDSW Preconditioner
    • References

16. Non-Geometric Convergence of the Classical Alternating Schwarz Method

    • Introduction
    • Domain Decomposition with Non-Lipschitz Overlap
    • ‘Slow’ Convergence in the Abstract Framework of the APM
    • The Dependence of the Convergence Rate on the Initial Function
    • Numerical Experiments
    • References

17. Global-in-Time Domain Decomposition for a Nonlinear Diffusion Problem

    • Introduction
    • Space-Time Domain Decomposition at the Continuous Level
    • Semi-Discrete Space-Time Domain Decomposition with Different Time Steps in the Subdomains
    • Numerical Experiment
    • References

18. A Two-Level Overlapping Schwarz Method Using Energy Minimizing Multiscale Finite Element Functions

    • Introduction
    • Multiscale Finite Element Basis Functions
    • Two-Level Overlapping Schwarz Algorithm
    • Numerical Results
    • References

19. Machine Learning in Adaptive FETI-DP – A Comparison of Smart and Random Training Data

    • Introduction
    • Model Problem and Adaptive FETI-DP
    • Machine Learning for Adaptive FETI-DP
    • Numerical Results
    • References

20. Nonoverlapping Additive Schwarz Method for hp-DGFEM with Higher-Order Penalty Terms

    • Introduction
    • High-Order Penalty h-p Discontinuous Galerkin Discretization
    • Nonoverlapping Additive Schwarz Method
    • Numerical Experiments
    • References

Part III: Contributed Talks and Posters (CT)

1. Multi-Step Variant of the Parareal Algorithm

   • Introduction
   • Parareal Algorithm and the Multi-Step Variant
   • Numerical Results
     • Dahlquist Equation
     • Advection-Diffusion Equation
     • Application to the Cathare Code
   • Conclusion
   • References

2. A Domain Decomposition Method for a Geological Crack

   • Introduction
   • Model Description
   • Domain Decomposition
   • Finite Element Discretization and the Algebraic Problem
   • Numerical Results
   • Conclusion
   • References

3. Fictitious Domain Method for an Inverse Problem in Volcanoes

   • General Framework and Problem Setting
   • Domain Decomposition: The Direct Solver
   • The Crack Inverse Problem
   • Numerical Experiments
   • Conclusion
   • References

4. A Schwarz Method for the Magnetotelluric Approximation of Maxwell’s Equations

   • Introduction
   • Well-Posedness, Schwarz Method, and Convergence
   • Numerical Examples
   • Conclusion
   • References

5. Can Classical Schwarz Methods for Time-Harmonic Elastic Waves Converge?

   • Mathematical Model
   • Classical Schwarz Algorithm
   • Numerical Experiments
   • Conclusion
   • References

6. Asymptotic Analysis for the Coupling Between Subdomains in Discrete Fracture Matrix Models

   • Introduction
   • Model Problem
   • Derivation of Reduced Models by Fourier Analysis
   • Comparison to the Literature
   • Numerical Results
   • Conclusion
   • References

7. A Nonlinear Elimination Preconditioned Inexact Newton Algorithm for Steady-State Incompressible Flow

   • Introduction
   • The Nonlinear Elimination Preconditioned Inexact Newton Algorithm
   • Numerical Experiments
   • Conclusions
   • References

8. A Neumann-Neumann Method for Anisotropic TDNNS Finite Elements in 3D Linear Elasticity

   • Introduction
   • TDNNS Formulation
   • Domain Decomposition
   • Neumann-Neumann Preconditioner
   • Numerical Experiments
   • Conclusion and Outlook
   • References

9. Domain Decomposition for the Closest Point Method

   • Introduction
   • The Closest Point Method
   • (Optimized) Restricted Additive Schwarz
   • Subdomain Construction
   • Transmission Conditions
   • Results
   • Conclusion
   • References

10. Towards a Time Adaptive Neumann-Neumann Waveform Relaxation Method for Thermal Fluid-Structure Interaction

    • Introduction
    • Model Problem
    • The Neumann-Neumann Waveform Relaxation Algorithm
    • Time Adaptive Method
    • Numerical Results
    • Conclusions and Further Work
    • References

11. Localization of Nonlinearities and Recycling in Dual Domain Decomposition

    • Introduction
    • Localized Nonlinearities in Dual Domain Decomposition
    • Model Problem and Nonlinear Solution Strategy
    • Localization of Nonlinearities
    • Adaptive Multipreconditioning
    • Recycling Methods for Dual Solutions
    • Numerical Results
    • Conclusions
    • References

12. New Coarse Corrections for Optimized Restricted Additive Schwarz Using PETSc

    • Introduction
    • Coarse Correction and the Two-Level RAS Method
    • Optimized Interface Conditions and the ORAS2 Method
    • Weak Scalability Results
    • Conclusions
    • References

13. On the Derivation of Optimized Transmission Conditions for the Stokes-Darcy Coupling

    • Introduction
    • The Stokes-Darcy Model
    • Numerical Study of the Optimized Schwarz Method
    • Conclusions
    • References