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<p>In this paper, we propose a linear and monolithic finite element method for the approximation of an incompressible viscous fluid interacting with an elastic and deforming plate. We use the arbitrary Lagrangian–Eulerian (ALE) approach that works in the reference domain, meaning that no re-meshing is needed during the numerical simulation. For time discretization, we employ the backward Euler method. For space discretization, we respectively use P1-bubble, P1, and P1 finite elements for the approximation of the fluid velocity, pressure, and structure displacement. We show that our method fulfills the geometrical conservation law and dissipates the total energy on the discrete level. Moreover, we prove the (optimal) linear convergence with respect to the sizes of the time step <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="211_2025_1461_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="11" /> </InlineMediaObject> <EquationSource Format="TEX">\(\tau \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>τ</mi> </math></EquationSource> </InlineEquation> and the mesh <i>h</i>. We present numerical experiments involving a substantially deforming fluid domain that do validate our theoretical results. A comparison with a fully implicit (thus nonlinear) scheme indicates that our semi-implicit linear scheme is faster and as accurate as the fully implicit one, at least in stable configurations.</p>

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Stability and error estimates of a linear numerical scheme approximating nonlinear fluid-structure interactions

  • 期刊论文

  • 发表日期：

    2025年3月24日

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  • Sebastian Schwarzacher,
  • Bangwei She,
  • Karel Tůma

摘要

In this paper, we propose a linear and monolithic finite element method for the approximation of an incompressible viscous fluid interacting with an elastic and deforming plate. We use the arbitrary Lagrangian–Eulerian (ALE) approach that works in the reference domain, meaning that no re-meshing is needed during the numerical simulation. For time discretization, we employ the backward Euler method. For space discretization, we respectively use P1-bubble, P1, and P1 finite elements for the approximation of the fluid velocity, pressure, and structure displacement. We show that our method fulfills the geometrical conservation law and dissipates the total energy on the discrete level. Moreover, we prove the (optimal) linear convergence with respect to the sizes of the time step \(\tau \) τ and the mesh h. We present numerical experiments involving a substantially deforming fluid domain that do validate our theoretical results. A comparison with a fully implicit (thus nonlinear) scheme indicates that our semi-implicit linear scheme is faster and as accurate as the fully implicit one, at least in stable configurations.