An efficient nonlinear multigrid method for a mixed finite element method of the Darcy-Forchheimer model is constructed in this paper. A Peaceman-Rachford type iteration is used as a smoother to decouple the nonlinearity from the divergence constraint. The nonlinear equation can be solved element-wise with a closed formulae. The linear saddle point system for the constraint is reduced into a symmetric positive definite system of Poisson type. Furthermore an empirical choice of the parameter used in the splitting is proposed and the resulting multigrid method is robust to the so-called Forchheimer number which controls the strength of the nonlinearity. By comparing the number of iterations and CPU time of different solvers in several numerical experiments, our multigrid method is shown to convergent with a rate independent of the mesh size and the Forchheimer number and with a nearly linear computational cost.

中文翻译：

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Darcy-Forchheimer模型的混合有限元方法的多重网格方法。

构造了一种有效的非线性多重网格方法，用于Darcy-Forchheimer模型的混合有限元方法。使用Peaceman-Rachford类型的迭代作为平滑器将非线性与发散约束解耦。非线性方程可以用封闭公式逐元素求解。用于约束的线性鞍点系统被简化为泊松型对称正定系统。此外，提出了在分割中使用的参数的经验选择，并且所得的多重网格方法对于控制非线性强度的所谓的Forchheimer数是鲁棒的。通过比较几个数值实验中不同求解器的迭代次数和CPU时间，