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<p>We present a monolithic divergence-conforming virtual element method for the linear fluid–structure interaction (FSI) problem with a thick structure. This problem consists of the time-dependent Stokes equations in the fluid domain coupled with the linear elastodynamics equations in the solid region. To keep the divergence be conforming, the global velocity is discretized by the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\textbf{H}(\textrm{div})\)</EquationSource> </InlineEquation> virtual element with an additional polynomial space defined on the element edges to approximate the tangential trace of the velocity, while the pressure is discretized by the discontinuous piecewise polynomials. Such spatial approximation combined with the Crank-Nicolson technique for the temporal discretization leads to a fully discrete scheme that produces an exactly divergence-free fluid velocity approximation. The a priori error estimates of the fully discrete scheme are well established, where the error bound is robust to the physical parameters. Finally, numerical examples are provided to verify that our method is not affected by the added mass effect and is robust to physical parameters.</p>

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Divergence-conforming virtual element method for the incompressible fluid–structure interaction model

  • 期刊论文

  • 发表日期：

    2026年8月4日

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  • Gang Wang,
  • Yaolong Jing,
  • Yinnian He

摘要

We present a monolithic divergence-conforming virtual element method for the linear fluid–structure interaction (FSI) problem with a thick structure. This problem consists of the time-dependent Stokes equations in the fluid domain coupled with the linear elastodynamics equations in the solid region. To keep the divergence be conforming, the global velocity is discretized by the \(\textbf{H}(\textrm{div})\) virtual element with an additional polynomial space defined on the element edges to approximate the tangential trace of the velocity, while the pressure is discretized by the discontinuous piecewise polynomials. Such spatial approximation combined with the Crank-Nicolson technique for the temporal discretization leads to a fully discrete scheme that produces an exactly divergence-free fluid velocity approximation. The a priori error estimates of the fully discrete scheme are well established, where the error bound is robust to the physical parameters. Finally, numerical examples are provided to verify that our method is not affected by the added mass effect and is robust to physical parameters.