Efficient Realization of the Mixed Finite Element Discretization for nonlinear Problems

TL;DR

This paper explores efficient implementation strategies for mixed finite element methods applied to nonlinear problems, focusing on hybridization, unknown reduction, and solution techniques, with applications to Darcy-Forchheimer flow in porous media.

Contribution

It establishes the equivalence between hybridized mixed finite element methods and nonconforming methods, and proposes practical solution approaches for nonlinear systems.

Findings

Hybridized formulation is equivalent to a nonconforming finite element method.

Static condensation reduces the number of unknowns effectively.

Newton's method is suitable for solving local and global nonlinear systems.

Abstract

We consider implementational aspects of the mixed finite element method for a special class of nonlinear problems. We establish the equivalence of the hybridized formulation of the mixed finite element method to a nonconforming finite element method with augmented Crouzeix-Raviart ansatz space. We discuss the reduction of unknowns by static condensation and propose Newton's method for the solution of local and global systems. Finally, we show, how such a nonlinear problem arises from the mixed formulation of Darcy-Forchheimer flow in porous media.