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.

翻译：用于双相位流和混合分流速度压力饱和度配方的非线性多格求解器。 提议了在全近近似方案(FAS)框架下的求解器, 扩展了我们以前关于不同扩散问题的非线性多格工作。 多格结构中的粗略空间是通过先汇总自由度, 然后解决某些本地流动问题构建的。 混合配方和选择粗略空间, 使我们能够在解答阶段中收集粗糙的问题, 而不必访问对多格方法的可伸缩性至关重要的细度。 具体地说, 上风通量的自然一般化可以直接在粗略水平上评估, 使用预合成的粗略通量基矢量。 由此产生的求解器适用于一般非结构化电网上的问题。 与标准单级牛顿方法相比,拟议的非线性多格求解器的性能通过具有挑战性的数字示例来证明。 观察到, 拟议的求解器对于高度非线性的问题和明显超越了多格的牛顿- 富尔基公司数字法例中, 。