A posteriori error analysis of a mixed FEM for the coupled Brinkman-Forchheimer/Darcy problem

TL;DR

This paper develops a reliable and efficient residual-based a posteriori error estimator for a mixed finite element scheme solving coupled Brinkman-Forchheimer and Darcy equations, with numerical validation including flow in heterogeneous media.

Contribution

It introduces the first a posteriori error estimator for the 2D mixed FEM for the coupled Brinkman-Forchheimer/Darcy problem, addressing nonlinear challenges.

Findings

Estimator is reliable and efficient based on theoretical analysis.

Numerical experiments confirm estimator accuracy and adaptive algorithm performance.

Flow in heterogeneous porous media demonstrates practical applicability.

Abstract

We consider a mixed variational formulation recently proposed for the coupling of the Brinkman--Forchheimer and Darcy equations and develop the first reliable and efficient residual-based a posteriori error estimator for the 2D version of the associated conforming mixed finite element scheme. For the reliability analysis, due to the nonlinear nature of the problem, we make use of the inf-sup condition and the strong monotonicity of the operators involved, along with a stable Helmholtz decomposition in Hilbert spaces and local approximation properties of the Raviart--Thomas and Cl\'ement interpolants. On the other hand, inverse inequalities, the localization technique through bubble functions, and known results from previous works are the main tools yielding the efficiency estimate. Finally, several numerical examples confirming the theoretical properties of the estimator and illustrating the performance of the associated adaptive algorithms are reported. In particular, the case of flow through a heterogeneous porous medium is considered.