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<p>A mechanical model and finite element method for the <i>simultaneous</i> solution of Stokes and incompressible Navier–Stokes flows on multiple curved surfaces over a bulk domain are proposed. The two-dimensional surfaces are defined implicitly by all level sets of a scalar function, bounded by the three-dimensional bulk domain. This bulk domain is discretized with hexahedral finite elements which do not necessarily conform with the level sets but with the boundary.The resulting numerical method is a hybrid between conforming and non-conforming finite element methods. Taylor–Hood elements or equal-order element pairs for velocity and pressure, together with stabilization techniques, are applied to fulfil the inf-sup conditions resulting from the mixed-type formulation of the governing equations. Numerical studies confirm good agreement with independently obtained solutions on selected, individual surfaces. Furthermore, higher-order convergence rates are obtained for sufficiently smooth solutions.</p>

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Simultaneous solution of incompressible Navier–Stokes flows on multiple surfaces

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  • 出版模式：

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  • 发表日期：

    2025年9月25日

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  • Michael Wolfgang Kaiser,
  • Thomas-Peter Fries

摘要

A mechanical model and finite element method for the simultaneous solution of Stokes and incompressible Navier–Stokes flows on multiple curved surfaces over a bulk domain are proposed. The two-dimensional surfaces are defined implicitly by all level sets of a scalar function, bounded by the three-dimensional bulk domain. This bulk domain is discretized with hexahedral finite elements which do not necessarily conform with the level sets but with the boundary.The resulting numerical method is a hybrid between conforming and non-conforming finite element methods. Taylor–Hood elements or equal-order element pairs for velocity and pressure, together with stabilization techniques, are applied to fulfil the inf-sup conditions resulting from the mixed-type formulation of the governing equations. Numerical studies confirm good agreement with independently obtained solutions on selected, individual surfaces. Furthermore, higher-order convergence rates are obtained for sufficiently smooth solutions.