A scalable high-order multigrid-FFT Poisson solver for unbounded domains on adaptive multiresolution grids

TL;DR

This work presents a scalable Poisson solver for unbounded domains implemented on adaptive multiresolution grids by coupling a multigrid V-cycle with a FFT-based coarse-grid solver. It achieves high-order accuracy using compact Mehrstellen stencils and ensures compatibility between lattice Green's functions and the discrete operator to handle unbounded directions. Validation on periodic and unbounded domains demonstrates both accuracy and scalability, with strong weak scaling observed on European HPC platforms. The approach enables efficient resolution of multi-scale Poisson problems common in CFD and electromagnetism, while maintaining load balance through a coarse-grid direct solve. Future work includes relaxing refinement constraints and porting to heterogeneous CPU-GPU architectures.

Abstract

Multigrid solvers are among the most efficient methods for solving the Poisson equation, which is ubiquitous in computational physics. For example, in the context of incompressible flows, it is typically the costliest operation. The present document expounds upon the implementation of a flexible multigrid solver that is capable of handling any type of boundary conditions within murphy, a multiresolution framework for solving partial differential equations (PDEs) on collocated adaptive grids. The utilization of a Fourier-based direct solver facilitates the attainment of flexibility and enhanced performance by accommodating any combination of unbounded and semi-unbounded boundary conditions. The employment of high-order compact stencils contributes to the reduction of communication demands while concurrently enhancing the accuracy of the system. The resulting solver is validated against analytical solutions for periodic and unbounded domains. In conclusion, the solver has been demonstrated to demonstrate scalability to 16,384 cores within the context of leading European high-performance computing infrastructures.

Figures (12)

• Figure 1: Global AMR grid and its corresponding uniform grid hierarchy
• Figure 1: Verification in a periodic unit cube domain for $\sigma = 0.01$ in \ref{['eq:per_uref']}: error behavior $E_\infty$ as a function of the refinement tolerance $\epsilon_r$ (left) and as a function of the resulting number of blocks (right); dashed lines show a linear behavior $E_\infty \propto \epsilon_r$ (left) and the powers $E_\infty \propto (\#blocks)^{-M/3}$ for $M=2,\,4,\,6$ (right, from top to bottom).
• Figure 1: Performance: weak scaling for fourth order stencil ($60$ blocks or $829440$ grid points per rank)
• Figure 2: Adaptive multigrid V-cycle applied to the grid depicted on \ref{['fig:grid_hierarchy']}
• Figure 2: Verification in an unbounded unit cube domain: error behavior $E_\infty$ as a function of the refinement tolerance $\epsilon_r$ (left) and as a function of the resulting number of blocks (right); dashed lines show a linear behavior $E_\infty \propto \epsilon_r$ (left) and the powers $E_\infty \propto (\#blocks)^{M/3}$ for $M=2,\,4,\,6$ (right, from top to bottom).