Multigrid Methods for A Mixed Finite Element Method of The Darcy-Forchheimer Model

TL;DR

This paper introduces an efficient nonlinear multigrid method for a mixed finite element approach to the Darcy-Forchheimer model, demonstrating robustness and near-linear computational cost across various parameters.

Contribution

It develops a novel multigrid algorithm with a Peaceman-Rachford smoother that effectively handles nonlinearity and divergence constraints in the Darcy-Forchheimer model.

Findings

Converges independently of mesh size and Forchheimer number

Achieves nearly linear computational cost

Robust to nonlinearity strength

Abstract

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.