Numerical Analysis of a Cut Finite Element Approach for Fully Eulerian Fluid-Structure Interaction with Fixed Interface

TL;DR

This paper presents a stable and accurate cut finite element method for linear fluid-structure interaction problems with a fixed interface, providing theoretical error estimates and numerical validation.

Contribution

It introduces a variational-monolithic unfitted finite element formulation with ghost penalty stabilization for FSI in Eulerian coordinates, including stability and optimal error analysis.

Findings

Proves optimal-order error estimates in space and time.

Demonstrates robustness of the method regardless of interface position.

Validates theoretical results with numerical experiments.

Abstract

This work develops and analyzes a variational-monolithic unfitted finite element formulation of a linear fluid-structure interaction problem in Eulerian coordinates with a fixed interface. The overall discretization is based on a backward Euler scheme in time and finite elements in space. For the spatial discretization we employ a cut finite element method on a mesh consisting of quadrilateral elements. We use a first-order in time formulation of the elasticity equations, inf-sup stable finite elements in the fluid part and Nitsche's method to incorporate the coupling conditions. Ghost penalty terms guarantee the robustness of the approach independently of the way the interface cuts the finite element mesh. The main objective is to establish stability and a priori error estimates. We prove optimal-order error estimates in space and time and substantiate them with numerical tests.