Stability Analysis of Monolithic Globally Divergence-Free ALE-HDG Methods for Fluid-Structure Interaction

TL;DR

This paper introduces two fully discrete monolithic finite element methods for fluid-structure interaction using a novel ALE mapping, ensuring stability and divergence-free velocity approximations, verified through numerical experiments.

Contribution

The paper presents new monolithic ALE-HDG methods with stability analysis and divergence-free velocities for FSI, combining hybridizable discontinuous Galerkin and continuous Galerkin techniques.

Findings

Stability results established for both semi-discrete and fully discrete schemes.

Velocity approximations are proven to be globally divergence-free.

Numerical experiments confirm the effectiveness of the proposed methods.

Abstract

In this paper, we propose two monolithic fully discrete finite element methods for fluid-structure interaction (FSI) based on a novel Piola-type Arbitrary Lagrangian-Eulerian (ALE) mapping. For the temporal discretization, we apply the backward Euler method to both the non-conservative and conservative formulations. For the spatial discretization, we adopt arbitrary order hybridizable discontinuous Galerkin (HDG) methods for the incompressible Navier-Stokes and linear elasticity equations, and a continuous Galerkin (CG) method for the fluid mesh movement. We derive stability results for both the temporal semi-discretization and the fully discretization, and show that the velocity approximations of the fully discrete schemes are globally divergence-free. Several numerical experiments are performed to verify the performance of the proposed methods.