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.

翻译：我们提出了一种结合代数一致粗化的无矩阵GPU多重网格预条件器，用于求解非规则域自适应八叉树网格上的泊松方程。在均匀分辨率区域内，粗化过程满足伽辽金原理。针对细化层级间的T型节点，我们提出了一种通量一致的粗网格校正方法，该方法在保持紧凑无矩阵表示的同时恢复了跨层级一致性。粗化算子采用适合GPU并行执行的紧凑无矩阵形式存储。数值实验表明，该方法具有二阶精度，与PCG联用时可实现网格无关收敛，并在流体模拟中的切割网格问题上展现了稳健性能。在单块NVIDIA RTX 4090 GPU上，该求解器在解析泊松测试中达到每秒超过2亿个单元的全求解吞吐量，在流体模拟压力投影问题中达到每秒超过7000万个单元。