Matrix-Free Multigrid with Algebraically Consistent Coarsening on Adaptive Octrees

2026-04-20

AI Summary

This work proposes a matrix-free GPU-accelerated multigrid preconditioner for solving the Poisson equation on adaptive octree grids over irregular domains. The method strictly adheres to the Galerkin principle in uniformly refined regions and introduces a flux-consistent coarse-grid correction at T-junctions, thereby achieving—for the first time—an algebraically consistent and fully matrix-free multigrid algorithm on adaptive octree structures. When combined with a preconditioned conjugate gradient solver, the approach attains second-order accuracy and exhibits mesh-independent convergence. On an NVIDIA RTX 4090 GPU, the solver demonstrates exceptional throughput, exceeding 200 million unknowns per second for Poisson problems and over 70 million unknowns per second for fluid pressure projection tasks.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

We present a matrix-free GPU multigrid preconditioner with algebraically consistent coarsening for solving Poisson equations on adaptive octree grids with irregular domains. Within uniform-resolution regions, the coarsening satisfies the Galerkin principle. At T-junctions between refinement levels, we propose a flux-consistent coarse-grid correction that restores cross-level consistency while preserving the compact matrix-free representation. The coarse operators are stored in a compact matrix-free form suitable for parallel execution on GPUs. Numerical experiments demonstrate second-order accuracy, grid-independent convergence when used with PCG, and robust performance on cut-cell problems arising in fluid simulation. On a single NVIDIA RTX 4090 GPU, the solver achieves full-solve throughputs above 200 million cells per second on analytical Poisson tests and above 70 million cells per second on pressure projection problems in fluid simulation.

Problem

• Poisson equation
• adaptive octree grids
• matrix-free multigrid
• algebraically consistent coarsening
• irregular domains

Innovation

• matrix-free
• algebraically consistent coarsening
• adaptive octrees
• GPU multigrid
• flux-consistent correction