A mixed FEM for the coupled Brinkman-Forchheimer/Darcy problem

TL;DR

This paper presents a mixed finite element method for modeling fluid flow through porous media governed by Brinkman--Forchheimer and Darcy equations, with stability, convergence analysis, and numerical validation.

Contribution

It introduces a novel mixed FEM formulation combining different elements and Lagrange multipliers for coupled Brinkman--Forchheimer and Darcy problems, with rigorous analysis.

Findings

Proved stability and convergence of the proposed scheme.

Derived a priori error estimates for the discretization.

Numerical tests confirm theoretical predictions.

Abstract

This paper develops the a priori analysis of a mixed finite element method for the filtration of an incompressible fluid through a non-deformable saturated porous medium with heterogeneous permeability. Flows are governed by the Brinkman--Forchheimer and Darcy equations in the more and less permeable regions, respectively, and the corresponding transmission conditions are given by mass conservation and continuity of momentum. We consider the standard mixed formulation in the Brinkman--Forchheimer domain and the dual-mixed one in the Darcy region, and we impose the continuity of the normal velocities by introducing suitable Lagrange multiplier. The finite element discretization involves Bernardi--Raugel and Raviart--Thomas elements for the velocities, piecewise constants for the pressures, and continuous piecewise linear elements for the Lagrange multiplier. Stability, convergence, and a priori error estimates for the associated Galerkin scheme are obtained. Numerical tests illustrate the theoretical results.