Convergence proof of the velocity field for a Stokes flow immersed boundary method.

*(English)*Zbl 1171.76042The author has performed convergence analysis for a stationary immersed boundary problem and for a small-amplitude dynamic problem. The immerse boundary method is a computational approach to model and simulate flow problems in which immersed elastic structures interact with fluid. Here considered is the standard model that describes a one-dimensional filament with prescribed force distribution immersed in a two-dimensional periodic fluid domain. The convergence of the velocity field is proved assuming stationary Stokes flow of incompressible Newtonian fluid. Spectral method is used to discretize the continuous Stokes problem expressed via the corresponding Green’s function for spatially periodic flows. Finite difference grid for the fluid domain, Lagrangian curvilinear mesh for the elastic material, and discrete delta function are used for communicating between the two grids. The error estimates have been tested against computational experiments.

Reviewer: Tomislav Zlatanovski (Skopje)

##### MSC:

76M22 | Spectral methods applied to problems in fluid mechanics |

76M20 | Finite difference methods applied to problems in fluid mechanics |

76D07 | Stokes and related (Oseen, etc.) flows |

74F10 | Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) |

65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |

65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |

##### Keywords:

Green’s function; discrete delta function; spectral discretization; finite difference method
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\textit{Y. Mori}, Commun. Pure Appl. Math. 61, No. 9, 1213--1263 (2008; Zbl 1171.76042)

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