CoarseSolvers for Exascale Solution of Poisson Problems

2026-06-18

AI Summary

This work addresses the coarse-grid efficiency bottleneck in solving the pressure Poisson problem arising from large-scale incompressible Navier–Stokes simulations on exascale supercomputers. To overcome this challenge, the authors propose a novel two-level Schwarz method to replace conventional algebraic multigrid (AMG) solvers as the coarse-grid solver within a p-multigrid preconditioner. The method constructs a structured yet non-nested global coarse space that enables communication-free interpolation between the original p-multigrid coarse space and the new coarse representation, substantially enhancing parallel scalability. Implemented within the Nek5000/RS framework using spectral/finite element discretizations, the approach is evaluated on the Summit and Frontier supercomputers, demonstrating superior strong scaling and solution efficiency compared to state-of-the-art AMG solvers such as BoomerAMG.

Abstract

WepresentatwolevelSchwarzmethodasanalternativetoAlgebraicMultigridmethod(AMG) used as the last level (coarse) solver of the p-multigrid pMG preconditioner for pressure Poission equation resulting from Spectral/Finite element descretization of incompressible Navier-Stokes eqaution. 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.

Problem

• Poisson equation
• coarse solver
• p-multigrid
• incompressible Navier-Stokes
• exascale

Innovation

• two-level Schwarz method
• non-nested coarse space
• communication-free interpolation
• p-multigrid preconditioner
• Poisson solver