Recent advances in domain decomposition methods for large-scale saddle point problems

Nataf, Frédéric; Tournier, Pierre-Henri

doi:10.5802/crmeca.130

Cited by 2 publications

(2 citation statements)

References 48 publications

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“…• FETI method [16,40] • P.L. Lions algorithm [20] • Inexact Coarse solve [30] • Saddle point problem [31] • Boundary Element Methods [29] • Multiscale Finite Element methods [27] • Time dependent Maxwell system [5] • Least Square problems [9] • Purely algebraic settings [9,17] Note that for Helmholtz or frequency Maxwell type problems, efficient coarse spaces are more easily built when they are based on a coarse grid discretisation of the underlying variational form as it is classically done in multigrid methods, see [18] and references therein for a detailed mathematical analysis and [4] for extensive numerical tests of various approaches.…”

Section: Coarse Space Constructionsmentioning

confidence: 99%

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Recent Advances in Adaptive Coarse Spaces and Availability in Open Source Libraries

Dolean¹,

Hecht²,

Jolivet³

et al. 2022

MathematicS in Action

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Section: Coarse Space Constructionsmentioning

confidence: 99%

“…In [31], we solve three dimensional elasticity problems for steel-rubber (ν = 0, 4999) structures (see Figure 5.1) discretized by a finite element method with continuous pressure with up to a billion degrees of freedom on 16,800 cores.…”

Section: Libraries For Large Scale Computationsmentioning

confidence: 99%

Recent Advances in Adaptive Coarse Spaces and Availability in Open Source Libraries

Dolean¹,

Hecht²,

Jolivet³

et al. 2022

MathematicS in Action

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Exploiting mesh structure to improve multigrid performance for saddle-point problems

Spies,

Olson,

MacLachlan

2024

The International Journal of High Performance Computing Applica

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In recent years, solvers for finite-element discretizations of linear or linearized saddle-point problems, like the Stokes and Oseen equations, have become well established. There are two main classes of preconditioners for such systems: those based on a block-factorization approach and those based on monolithic multigrid. Both classes of preconditioners have several critical choices to be made in their composition, such as the selection of a suitable relaxation scheme for monolithic multigrid. From existing studies, some insight can be gained as to what options are preferable in low-performance computing settings, but there are very few fair comparisons of these approaches in the literature, particularly for modern architectures, such as GPUs. In this paper, we perform a comparison between a Block-Triangular preconditioner and monolithic multigrid methods with the three most common choices of relaxation scheme – Braess-Sarazin, Vanka, and Schur-Uzawa. We develop a performant Vanka relaxation algorithm for structured-grid discretizations, which takes advantage of memory efficiencies in this setting. We detail the behavior of the various CUDA kernels for the multigrid relaxation schemes and evaluate their individual arithmetic intensity, performance, and runtime. Running a preconditioned FGMRES solver for the Stokes equations with these preconditioners allows us to compare their efficiency in a practical setting. We show that monolithic multigrid can outperform Block-Triangular preconditioning, and that using Vanka or Braess-Sarazin relaxation is most efficient. Even though multigrid with Vanka relaxation exhibits reduced performance on the CPU (up to 100% slower than Braess-Sarazin), it is able to outperform Braess-Sarazin by more than 20% on the GPU, making it a competitive algorithm, especially given the high amount of algorithmic tuning needed for effective Braess-Sarazin relaxation.

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Recent advances in domain decomposition methods for large-scale saddle point problems

Cited by 2 publications

References 48 publications

Recent Advances in Adaptive Coarse Spaces and Availability in Open Source Libraries

Recent Advances in Adaptive Coarse Spaces and Availability in Open Source Libraries

Exploiting mesh structure to improve multigrid performance for saddle-point problems

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