This paper presents a matrix-free multigrid method for solving the Stokes problem, discretized using $H^{\text{div}}$-conforming discontinuous Galerkin methods. We employ a Schur complement method combined with the fast diagonalization method for the efficient evaluation of the local solver within the multiplicative Schwarz smoother. This approach operates directly on both the velocity and pressure spaces, eliminating the need for a global Schur complement approximation. By leveraging the tensor product structure of Raviart-Thomas elements and an optimized, conflict-free shared memory access pattern, the matrix-free operator evaluation demonstrates excellent performance numbers, reaching over one billion degrees of freedom per second on a single NVIDIA A100 GPU. Numerical results indicate efficiency comparable to that of the three-dimensional Poisson problem.

翻译：本文提出了一种用于求解Stokes问题的无矩阵多重网格方法，该问题采用$H^{\text{div}}$协调间断伽辽金方法进行离散。我们采用Schur补方法与快速对角化方法相结合，以高效实现乘法型Schwarz光滑子中的局部求解器。该方法直接在速度和压力空间上操作，无需全局Schur补近似。通过利用Raviart-Thomas单元的张量积结构及优化的无冲突共享内存访问模式，无矩阵算子求值展现出卓越的性能指标，在单个NVIDIA A100 GPU上每秒可处理超过十亿自由度。数值结果表明其效率与三维Poisson问题相当。