Matrix-Free Multigrid with Algebraically Consistent Coarsening on Adaptive Octrees

TL;DR

This paper introduces a GPU-optimized, matrix-free multigrid preconditioner with algebraically consistent coarsening for adaptive octree grids, achieving high throughput and accuracy in fluid simulation problems.

Contribution

It develops a novel flux-consistent coarse-grid correction method for T-junctions that maintains the matrix-free structure and improves multigrid performance on adaptive grids.

Findings

Achieves second-order accuracy and grid-independent convergence.

Reaches over 200 million cells per second on analytical Poisson tests.

Performs robustly on cut-cell fluid simulation problems.

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.