Published

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

Jian Huang, Long Chen, Hongxing Rui

Journal of Scientific Computing. 74(1), 396–411, 2018.

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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.