pith. sign in

arxiv: 2606.20496 · v2 · pith:76JNAX3Rnew · submitted 2026-06-18 · 🧮 math.NA · cs.DC· cs.MS· cs.NA

Coarse Solvers for Exascale Solution of Poisson Problems

Thilina Ratnayaka , Paul Fischer , Luke Olson This is my paper

Pith reviewed 2026-06-26 16:10 UTC · model grok-4.3

classification 🧮 math.NA cs.DCcs.MScs.NA
keywords Schwarz methodp-multigridcoarse solverPoisson equationincompressible Navier-Stokesnon-nested coarse spaceexascale solver
0
0 comments X

The pith

A structured non-nested coarse space for the global problem in a two-level Schwarz method serves as an effective coarse solver for pressure Poisson equations in p-multigrid preconditioners.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a two-level Schwarz method as an alternative to algebraic multigrid for the final coarse solve inside p-multigrid preconditioners applied to the pressure Poisson equation from spectral-element discretizations of incompressible Navier-Stokes flow. The central object is a novel structured non-nested coarse space that defines the global coarse problem. Because the space is structured, the required interpolation between the original p-multigrid coarse space and the global coarse problem needs no inter-process messages. Experiments performed with the Nek5000/RS code on Summit and Frontier machines indicate that the resulting solver matches or exceeds the performance of the BoomerAMG implementation.

Core claim

We present a two-level Schwarz method consisting of a local problem on the original pMG coarse space together with a global coarse problem on a novel structured non-nested coarse space; the structure permits communication-free interpolation between the two spaces, and the overall method functions as an effective coarse solver inside the pMG preconditioner for the pressure Poisson equation.

What carries the argument

The structured non-nested coarse space for the global coarse problem, which supplies the global correction while enabling direct, communication-free transfer from the pMG coarse space.

If this is right

  • The two-level Schwarz method can replace AMG as the last-level solver inside pMG preconditioners for the pressure Poisson equation.
  • The communication-free interpolation removes a potential latency bottleneck at extreme processor counts.
  • The method was shown to be competitive with BoomerAMG on production-scale runs of Nek5000/RS on Summit and Frontier.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same structured-space construction could be tested on other elliptic operators that arise in fluid problems.
  • Communication-free coarse-grid transfer may become increasingly valuable once node counts exceed current exascale machines.
  • Because the space is non-nested, it may allow easier incorporation of locally refined regions without rebuilding the entire hierarchy.

Load-bearing premise

The non-nested structured coarse space preserves the approximation and stability properties needed for the two-level Schwarz method to act as an effective coarse solver.

What would settle it

A Poisson problem on which the two-level Schwarz preconditioner with the proposed coarse space either diverges or requires substantially more iterations than the corresponding AMG solver.

Figures

Figures reproduced from arXiv: 2606.20496 by Luke Olson, Paul Fischer, Thilina Ratnayaka.

Figure 1
Figure 1. Figure 1: Two-level Schwarz illustration. (a) Parallel partition (𝑃 = 2) of the 𝑁 = 1 mesh: each rank solves a local Poisson problem on their respective shaded region (green or blue), including a one- or two-element overlap extension, shown in gray. (b) Vertex-based support of the coarse grid interpolants: shown in blue is the support for Φ22. The red triangles indicate border elements, for which support of Ψ22 woul… view at source ↗
Figure 2
Figure 2. Figure 2: Reduced-space basis functions in 1D: (a) standard, following Eq. (9), showing overlapping support of Ψ𝑗−1 and Ψ𝑗+1; (b) gap-based support; (c) element-centroid-based support. The overlap in case (a) leads to a 5-point stencil while (b) and (c) yield a 3-point stencil such that 𝐴𝑟 is tridiagonal [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Test cases (left to right): T-junction (𝐸=62176) configuration, including coarse-space cells; 146-pebble (𝐸=62138) mesh illustrating an eight-way mesh partition; 45000-pebble (𝐸=13032440) mesh, and 352625-pebble (𝐸=98782067) mesh. T. Ratnayaka, P. Fischer, L. Olson: Preprint submitted to Elsevier Page 15 of 14 [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Nek5000 test results for different coarse solvers: (top) iteration count-per-step; (bottom) cumulative iteration counts [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: NekRS results for different coarse solvers: cumulative iteration counts. T. Ratnayaka, P. Fischer, L. Olson: Preprint submitted to Elsevier Page 16 of 14 [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Strong scaling study of 45000 pebbles mesh with NekRS. Left to right: Navier-Stokes, Pressure and Coarse grid solve time per timestep for AMG and two level Schwarz method; Parallel efficiency for AMG and two level Schwarz method; Breakdown of the coarse grid solve time for the two level Schwarz method [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Estimated solve times for Schwarz and AMG for different 𝛾 and 𝐿 values. T. Ratnayaka, P. Fischer, L. Olson: Preprint submitted to Elsevier Page 17 of 14 [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
read the original abstract

We present a two-level Schwarz method as an alternative to Algebraic Multigrid method(AMG) used as the last level (coarse) solver of the p-multigrid pMG preconditioner for pressure Poisson equation resulting from Spectral/Finite element descretization of incompressible Navier-Stokes equation. Proposed Schwarz method consits of a local problem in the original pMG coarse space and a global coarse problem. Main contribution of the paper is a novel, structured and a non-nested coarse space for the global coarse problem. Structured nature of the proposed global coarse space enable communication-free interpolation between the original p-multgrid coarse space and the global coarse problem. We demonstrate the effectiveness of the proposed method compared to the state of the art AMG solver BoomerAMG by a series of experiments performed using Nek5000/RS, a suite of highly scalable incompressible Navier-Stokes solvers, on Summit/Frontier supercomputers at Oak Ridge Leadership Computing Facility.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper proposes a two-level Schwarz method as an alternative coarse solver to AMG within a p-multigrid (pMG) preconditioner for the pressure Poisson equation arising from spectral/finite-element discretizations of incompressible Navier-Stokes equations. The central contribution is a novel structured, non-nested coarse space for the global coarse problem that enables communication-free interpolation to the original pMG coarse space. Effectiveness relative to BoomerAMG is asserted on the basis of experiments performed with Nek5000/RS on Summit and Frontier.

Significance. If the non-nested coarse space is shown to satisfy the stable-decomposition and approximation properties required for two-level Schwarz convergence, the method would supply a communication-avoiding coarse solver suitable for exascale pMG preconditioning of Poisson problems. The structured construction could reduce inter-node communication relative to standard AMG while remaining compatible with existing high-order CFD codes.

major comments (2)
  1. [Abstract] Abstract (method paragraph): the claim that the structured non-nested coarse space 'enable[s] communication-free interpolation' while still making the two-level Schwarz method an effective coarse solver rests on the unverified assumption that the space preserves the approximation and stability properties needed for a stable decomposition in the Schwarz theory; no analytic argument or numerical check of these properties is indicated.
  2. [Abstract] Abstract (final sentence): the assertion that experiments 'demonstrate the effectiveness' versus BoomerAMG supplies no quantitative metrics, iteration counts, convergence rates, or error tables, so it is impossible to determine whether the reported runs actually support the central claim that the new coarse solver is competitive.
minor comments (1)
  1. [Abstract] Abstract: 'descretization' should be 'discretization'; 'consits' should be 'consists'; 'p-multgrid' should be 'p-multigrid'.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. We address each major comment below and indicate the revisions we will make to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract (method paragraph): the claim that the structured non-nested coarse space 'enable[s] communication-free interpolation' while still making the two-level Schwarz method an effective coarse solver rests on the unverified assumption that the space preserves the approximation and stability properties needed for a stable decomposition in the Schwarz theory; no analytic argument or numerical check of these properties is indicated.

    Authors: We agree that an explicit verification of the approximation and stability properties would strengthen the presentation. The manuscript demonstrates effectiveness through large-scale numerical experiments on Summit and Frontier that implicitly rely on these properties holding in practice. To address the comment directly, we will add a short subsection in the methods or results section providing either a brief theoretical justification for why the structured non-nested construction preserves the required properties or additional numerical checks (e.g., stable-decomposition constants computed on representative meshes). revision: yes

  2. Referee: [Abstract] Abstract (final sentence): the assertion that experiments 'demonstrate the effectiveness' versus BoomerAMG supplies no quantitative metrics, iteration counts, convergence rates, or error tables, so it is impossible to determine whether the reported runs actually support the central claim that the new coarse solver is competitive.

    Authors: The abstract is length-constrained and therefore summarizes rather than quantifies. The full manuscript already contains the requested data in the form of iteration counts, wall-clock timings, strong-scaling plots, and direct comparisons against BoomerAMG across multiple problem sizes. We will revise the final sentence of the abstract to include one or two representative quantitative metrics (e.g., iteration counts and relative wall-time reduction) that support the effectiveness claim. revision: yes

Circularity Check

0 steps flagged

No circularity; novel non-nested coarse space is introduced and validated empirically against external solver.

full rationale

The paper introduces a new structured non-nested coarse space for the global problem in a two-level Schwarz method, with the central claim supported by experimental comparison to BoomerAMG on exascale hardware. No load-bearing step reduces by construction to a fitted parameter, self-citation chain, or renamed input; the method is presented as novel and the effectiveness is shown via independent benchmarks rather than analytic self-reference. The assumption that the space preserves required approximation/stability properties is stated but not derived from prior self-work in a circular manner.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no information on free parameters, background axioms, or new postulated entities; full manuscript required to populate the ledger.

pith-pipeline@v0.9.1-grok · 5694 in / 1108 out tokens · 49486 ms · 2026-06-26T16:10:20.494480+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

33 extracted references · 1 linked inside Pith

  1. [1]

    Highly parallel sparse triangular solution

    Alvarado, F.L., Pothen, A., Schreiber, R., 1992. Highly parallel sparse triangular solution. Pennsylvania State University, Department of Computer Science

    1992

  2. [2]

    Algorithm 837: Amd, an approximate minimum degree ordering algorithm

    Amestoy, P.R., Davis, T.A., Duff, I.S., 2004. Algorithm 837: Amd, an approximate minimum degree ordering algorithm. ACM Transactions on Mathematical Software (TOMS) 30, 381–388

    2004

  3. [3]

    Mfem: A modular finite element methods library

    Anderson,R.,Andrej,J.,Barker,A.,Bramwell,J.,Camier,J.S.,Cerveny,J.,Dobrev,V.,Dudouit,Y.,Fisher,A.,Kolev,T.,etal.,2021. Mfem: A modular finite element methods library. Computers & Mathematics with Applications 81, 42–74

    2021

  4. [4]

    Theanalysisofmultigridalgorithmswithnonnestedspacesornoninheritedquadraticforms

    Bramble,J.H.,Pasciak,J.E.,Xu,J.,1991. Theanalysisofmultigridalgorithmswithnonnestedspacesornoninheritedquadraticforms. Math. of Comp. 56, 1–34

    1991

  5. [5]

    The use of pointwise interpolation in domain decomposition methods with nonnested meshes

    Cai, X.C., 1995. The use of pointwise interpolation in domain decomposition methods with nonnested meshes. SIAM J. Sci. Comput. 16, 250–256

    1995

  6. [6]

    ArestrictedadditiveSchwarzpreconditionerforgeneralsparselinearsystems

    Cai,X.C.,Sarkis,M.,1999. ArestrictedadditiveSchwarzpreconditionerforgeneralsparselinearsystems. SIAMJ.Sci.Comput21,792–797

    1999

  7. [7]

    Overlapping schwarz methods on unstructured meshes using non-matching coarse grids

    Chan, T.F., Smith, B.F., Zou, J., 1996. Overlapping schwarz methods on unstructured meshes using non-matching coarse grids. Num. Math. 73, 149–167

    1996

  8. [8]

    Algorithm 887: Cholmod, supernodal sparse cholesky factorization and update/downdate

    Chen, Y., Davis, T.A., Hager, W.W., Rajamanickam, S., 2008. Algorithm 887: Cholmod, supernodal sparse cholesky factorization and update/downdate. ACM Transactions on Mathematical Software (TOMS) 35, 1–14

    2008

  9. [9]

    High-order methods for incompressible fluid flow

    Deville, M., Fischer, P., Mund, E., 2002. High-order methods for incompressible fluid flow. Cambridge University Press, Cambridge

    2002

  10. [10]

    An Additive Variant of the Schwarz Alternating Method for the Case of Many Subregions

    Dryja, M., Widlund, O., 1987. An Additive Variant of the Schwarz Alternating Method for the Case of Many Subregions. Technical Report TR 339. Courant Inst., NYU. Dept. Comp. Sci

    1987

  11. [11]

    An additive Schwarz algorithm for two- and three-dimensional finite element elliptic problems, in: Chan, T., Glowinski, R., Périaux, J., Widlund, O

    Dryja, M., Widlund, O., 1989. An additive Schwarz algorithm for two- and three-dimensional finite element elliptic problems, in: Chan, T., Glowinski, R., Périaux, J., Widlund, O. (Eds.), Domain Decomposition Methods, SIAM

    1989

  12. [12]

    Domain decomposition algorithms with small overlap

    Dryja, M., Widlund, O., 1994. Domain decomposition algorithms with small overlap. SIAM J. Sci. Comput. 15, 604–620

    1994

  13. [13]

    Tailoringdomaindecompositionmethodsforefficientparallelcoarsegridsolutionandforsystemswithmanyright hand sides

    Farhat,C.,Chen,P.,1994. Tailoringdomaindecompositionmethodsforefficientparallelcoarsegridsolutionandforsystemswithmanyright hand sides. Contemporary Mathematics 180, 401–406

    1994

  14. [14]

    Projection techniques for iterative solution of𝐴𝑥=𝑏 with successive right-hand sides

    Fischer, P., 1993. Projection techniques for iterative solution of𝐴𝑥=𝑏 with successive right-hand sides. Technical Report 93–90. ICASE. Hampton,Va

    1993

  15. [15]

    Parallelmulti-levelsolversforspectralelementmethods,in:Ilin,A.,Scott,L.(Eds.),ThirdInt.ConferenceonSpectraland High Order Methods, Houston J

    Fischer,P.,1996. Parallelmulti-levelsolversforspectralelementmethods,in:Ilin,A.,Scott,L.(Eds.),ThirdInt.ConferenceonSpectraland High Order Methods, Houston J. of Mathematics. pp. 595–604

    1996

  16. [16]

    Projection techniques for iterative solution of𝐴𝑥=𝑏 with successive right-hand sides

    Fischer, P., 1998. Projection techniques for iterative solution of𝐴𝑥=𝑏 with successive right-hand sides. Comput. Methods Appl. Mech. Engrg. 163, 193–204

    1998

  17. [17]

    Scaling limits for PDE-based simulation (invited), in: 22nd AIAA Computational Fluid Dynamics Conference, AIAA Aviation, AIAA 2015-3049

    Fischer, P., Heisey, K., Min, M., 2015. Scaling limits for PDE-based simulation (invited), in: 22nd AIAA Computational Fluid Dynamics Conference, AIAA Aviation, AIAA 2015-3049

    2015

  18. [18]

    HybridSchwarz-multigridmethodsforthespectralelementmethod:ExtensionstoNavier-Stokes,in:Kornhuber, R., Hoppe, R., Périaux, J., Pironneau, O., Widlund, O., Xu, J

    Fischer,P.,Lottes,J.,2004. HybridSchwarz-multigridmethodsforthespectralelementmethod:ExtensionstoNavier-Stokes,in:Kornhuber, R., Hoppe, R., Périaux, J., Pironneau, O., Widlund, O., Xu, J. (Eds.), Domain Decomposition Methods in Science and Engineering Series, Springer, Berlin

    2004

  19. [19]

    An overlapping Schwarz method for spectral element simulation of three-dimensional incompressible flows, in: Bjørstad, P., Luskin, M

    Fischer, P., Miller, N., Tufo, H., 2000. An overlapping Schwarz method for spectral element simulation of three-dimensional incompressible flows, in: Bjørstad, P., Luskin, M. (Eds.), Parallel Solution of Partial Differential Equations, Springer, Berlin. pp. 158–180

    2000

  20. [20]

    BoomerAMG: a parallel algebraic multigrid solver and preconditioner

    Henson, V., Yang, U., 2002. BoomerAMG: a parallel algebraic multigrid solver and preconditioner. Applied Numerical Mathematics 41, 155–177

    2002

  21. [21]

    Finite elements in computational electromagnetism

    Hiptmair, R., 2002. Finite elements in computational electromagnetism. Acta Numerica 11, 237–339

    2002

  22. [22]

    Hybrid multigrid/Schwarz algorithms for the spectral element method

    Lottes, J.W., Fischer, P.F., 2005. Hybrid multigrid/Schwarz algorithms for the spectral element method. J. Sci. Comput. 24, 45–78

    2005

  23. [23]

    OCCA: A unified approach to multi-threading languages

    Medina, D.S., St-Cyr, A., Warburton, T., 2014. OCCA: A unified approach to multi-threading languages. arXiv preprint arXiv:1403.0968 . T. Ratnayaka, P. Fischer, L. Olson:Preprint submitted to ElsevierPage 13 of 14 Fast Coarse Solvers

    Pith/arXiv arXiv 2014

  24. [24]

    A spectrally accurate method for overlapping grid solution of incompressible Navier-Stokes equations

    Merrill, B., Peet, Y., Fischer, P., Lottes, J., 2016. A spectrally accurate method for overlapping grid solution of incompressible Navier-Stokes equations. J. Comput. Phys. 307, 60–93

    2016

  25. [25]

    Min, M., Lan, Y., Fischer, P., Merzari, E., Kerkemeier, S., Phillips, M., Rathnayake, T., Novak, A., Gaston, D., Chalmers, N., Warburton, T.,

  26. [26]

    Optimization of full-core reactor simulations on Summit, in: Proc. Conf. on Supercomput., IEEE

  27. [27]

    Multiratetime-steppingfortheincompressibleNavier-Stokesequationsinoverlappinggrids

    Mittal,K.,Dutta,S.,Fischer,P.,2020. Multiratetime-steppingfortheincompressibleNavier-Stokesequationsinoverlappinggrids. J.Comput. Phys. 437, 110335

    2020

  28. [28]

    A spectral element method for fluid dynamics : laminar flow in a channel expansion

    Patera, A., 1984. A spectral element method for fluid dynamics : laminar flow in a channel expansion. J. Comput. Phys. 54, 468–488

    1984

  29. [29]

    TuningspectralelementpreconditionersforparallelscalabilityonGPUs,in:Proc.ofthe2022 SIAM Conf

    Phillips,M.,Kerkemeier,S.,Fischer,P.,2022. TuningspectralelementpreconditionersforparallelscalabilityonGPUs,in:Proc.ofthe2022 SIAM Conf. on Par. Proc. for Sci. Comp., SIAM. pp. 37–48

    2022

  30. [30]

    Partitioning sparse matrices with eigenvectors of graphs

    Pothen, A., Simon, H., Liou, K., 1990. Partitioning sparse matrices with eigenvectors of graphs. SIAM J. Matrix Anal. Appl. 11, 430–452

    1990

  31. [31]

    DomainDecomposition:ParallelMultilevelMethodsforEllipticPDEs

    Smith,B.,Bjørstad,P.,Gropp,W.,1996. DomainDecomposition:ParallelMultilevelMethodsforEllipticPDEs. CambridgeUniversityPress, Cambridge

    1996

  32. [32]

    Fast parallel direct solvers for coarse grid problems

    Tufo, H., Fischer, P., 2001. Fast parallel direct solvers for coarse grid problems. J. Parallel Distrib. Comput. 61, 151–177

    2001

  33. [33]

    The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids

    Xu, J., 1996. The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids. Computing 56, 215–235. T. Ratnayaka, P. Fischer, L. Olson:Preprint submitted to ElsevierPage 14 of 14 Fast Coarse Solvers Φ00 Φ30 Φ31 Φ32 Φ13 Φ21 Φ10 Φ02 Φ20 Φ23 Φ11 Φ03 Φ22 Φ01 Φ12 Φ33 Φ 33 Φ 02 Φ 22 Φ 01 Φ 23 Φ 30 Φ 20 Φ 03 Φ 12 Φ 10 Φ 21 Φ ...

    1996