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

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.