# Residual-based a posteriori error estimates for a conforming finite element discretization of the Navier-Stokes/Darcy coupled problem

## TL;DR

This paper introduces a new residual-based a posteriori error estimator for a conforming finite element method coupling Navier-Stokes and Darcy flows, proving its reliability and efficiency on isotropic meshes.

## Contribution

The paper presents a novel residual-based a posteriori error estimator for coupled Navier-Stokes/Darcy problems, with proven reliability and efficiency, applicable to various finite element subspaces.

## Key findings

- The error estimator is both reliable and efficient.
- The method is applicable to different finite element subspaces.
- The analysis extends to other stable Galerkin schemes.

## Abstract

We consider in this paper, a new a posteriori residual type error estimator of a conforming mixed finite element method for the coupling of fluid flow with porous media flow on isotropic meshes. Flows are governed by the Navier-Stokes and Darcy equations, respectively, and the corresponding transmission conditions are given by mass conservation, balance of normal forces, and the Beavers-Joseph-Saffman law. The finite element subspaces consider Bernardi-Raugel and Raviart-Thomas elements for the velocities, piecewise constants for the pressures, and continuous piecewise linear elements for a Lagrange multiplier defined on the interface. The posteriori error estimate is based on a suitable evaluation on the residual of the finite element solution. It is proven that the a posteriori error estimate provided in this paper is both reliable and efficient. In addition, our analysis can be extended to other finite element subspaces yielding a stable Galerkin scheme.