A Nitsche method for Navier--Stokes/generalized poroelasticity interface problems

TL;DR

This paper introduces a Nitsche-based finite element method for coupled Navier--Stokes and poroelastic flow problems, ensuring stability and convergence through a unified discretization and rigorous analysis.

Contribution

It develops a novel monolithic finite element scheme with Nitsche-type interface treatment for coupled flow problems, providing stability, error estimates, and numerical validation.

Findings

Method is stable and convergent with mesh-independent penalty parameters.

Numerical tests confirm theoretical convergence rates.

Accurately captures coupled fluid-structure dynamics.

Abstract

We consider a time-dependent coupled Navier--Stokes/generalized poroelastic flow problem and propose a unified and monolithic finite element discretization based on implicit time stepping. To handle the fluid-structure interface we employ a Nitsche-type formulation. The resulting discrete problem is shown to be well-posed using the theory of differential-algebraic equations (DAEs) and the Banach fixed-point theorem. We prove stability and derive a priori error estimates for the fully discrete scheme. The stability and convergence of the method are ensured by a properly chosen penalty parameter independent of the mesh size. Numerical tests are presented to confirm the theoretical convergence rates and to illustrate the ability of the method to capture the coupled dynamics accurately.