Matrix-Free Multigrid with Algebraically Consistent Coarsening on Adaptive Octrees
This work provides an efficient GPU solver for Poisson problems on adaptive grids with irregular domains, benefiting computational fluid dynamics and other simulation fields.
The paper presents a matrix-free GPU multigrid preconditioner for Poisson equations on adaptive octree grids, achieving second-order accuracy and grid-independent convergence. On an RTX 4090, it reaches over 200 million cells per second on analytical tests and over 70 million on fluid simulation pressure projections.
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.