We propose a robust, adaptive coarse-grid correction scheme for matrix-free geometric multigrid targeting PDEs with strongly varying coefficients. The method combines uniform geometric coarsening of the underlying grid with heterogeneous coarse-grid operators: Galerkin coarse grid approximation is applied locally in regions with large coefficient gradients, while lightweight, direct coarse grid approximation is used elsewhere. This selective application ensures that local Galerkin operators are computed and stored only where necessary, minimizing memory requirements while maintaining robust convergence. We demonstrate the method on a suite of sinker benchmark problems for the generalized Stokes equation, including grid-aligned and unaligned viscosity jumps, smoothly varying viscosity functions with large gradients, and different viscosity evaluation techniques. We analytically quantify the solver's memory consumption and demonstrate its efficiency by solving Stokes problems with $10^{10}$ degrees of freedom, viscosity jumps of $10^{6}$ magnitude, and more than 100{,}000 parallel processes.

翻译：本文针对具有强变化系数的偏微分方程，提出了一种鲁棒的自适应粗网格校正方案，适用于无矩阵几何多重网格。该方法将底层网格的均匀几何粗化与异构粗网格算子相结合：在系数梯度较大的区域局部应用Galerkin粗网格近似，而在其他区域使用轻量级直接粗网格近似。这种选择性应用确保仅在必要时计算和存储局部Galerkin算子，在保持鲁棒收敛性的同时最小化内存需求。我们通过一系列广义Stokes方程的沉陷基准问题验证该方法，包括网格对齐与非对齐的黏度跃变、具有大梯度的光滑变化黏度函数以及不同的黏度评估技术。我们通过解析量化求解器的内存消耗，并展示了其在求解具有$10^{10}$个自由度、$10^{6}$量级黏度跃变以及超过100,000个并行进程的Stokes问题时的效率。