Multigrid Methods for A Mixed Finite Element Method of The Darcy-Forchheimer Model
It provides a robust and efficient solver for a nonlinear flow problem in porous media, which is important for applications in geoscience and engineering.
This paper constructs an efficient nonlinear multigrid method for a mixed finite element discretization of the Darcy-Forchheimer model, achieving convergence rates independent of mesh size and Forchheimer number with nearly linear computational cost.
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.