# A new unified stabilized mixed finite element method of the Stokes-Darcy coupled problem: Isotropic discretization

## TL;DR

This paper introduces a new unified mixed finite element method for coupled Stokes-Darcy flows on isotropic meshes, providing stability, convergence analysis, and numerical validation for fluid-porous media interactions.

## Contribution

It develops a novel finite element approach combining a modified Darcy problem with a nonconforming element for the coupled flow, along with theoretical analysis and numerical validation.

## Key findings

- Method demonstrates excellent stability.
- Convergence analysis confirms accuracy.
- Numerical experiments validate theoretical results.

## Abstract

In this paper we develop an a priori error analysis of a new unified mixed finite element method for the coupling of fluid flow with porous media flow in $\mathbb{R}^N$, $N\in\{2,3\}$ on isotropic meshes. Flows are governed by the 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 approach utilizes a modification of the Darcy problem which allows us to apply a variant nonconforming Crouzeix-Raviart finite element to the whole coupled Stokes-Darcy problem. The well-posedness of the finite element scheme and its convergence analysis are derived. Finally, the numerical experiments are presented, which confirm the excellent stability and accuracy of our method.