A nonlinear multigrid solver for two-phase flow and transport in a mixed fractional-flow velocity-pressure-saturation formulation is proposed. The solver, which is under the framework of the full approximation scheme (FAS), extends our previous work on nonlinear multigrid for heterogeneous diffusion problems. The coarse spaces in the multigrid hierarchy are constructed by first aggregating degrees of freedom, and then solving some local flow problems. The mixed formulation and the choice of coarse spaces allow us to assemble the coarse problems without visiting finer levels during the solving phase, which is crucial for the scalability of multigrid methods. Specifically, a natural generalization of the upwind flux can be evaluated directly on coarse levels using the precomputed coarse flux basis vectors. The resulting solver is applicable to problems discretized on general unstructured grids. The performance of the proposed nonlinear multigrid solver in comparison with the standard single level Newton's method is demonstrated through challenging numerical examples. It is observed that the proposed solver is robust for highly nonlinear problems and clearly outperforms Newton's method in the case of high Courant-Friedrichs-Lewy (CFL) numbers.

中文翻译：

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多孔介质中两相流和输运的基于聚合的非线性多重网格求解器

提出了一种用于混合分数流速度-压力-饱和度公式中的两相流和输运的非线性多重网格求解器。求解器在完全逼近方案 (FAS) 的框架下，扩展了我们之前在非线性多重网格上针对异构扩散问题的工作。多重网格层次结构中的粗空间是通过首先聚合自由度，然后解决一些局部流问题来构建的。混合公式和粗空间的选择使我们能够在求解阶段无需访问更精细的级别即可组装粗问题，这对于多重网格方法的可扩展性至关重要。具体而言，可以使用预先计算的粗通量基础向量直接在粗水平上评估逆风通量的自然概括。所得求解器适用于在一般非结构化网格上离散化的问题。与标准单级牛顿法相比，所提出的非线性多重网格求解器的性能通过具有挑战性的数值例子来证明。据观察，所提出的求解器对于高度非线性问题具有鲁棒性，并且在高 Courant-Friedrichs-Lewy (CFL) 数的情况下明显优于牛顿方法。