*(English)*Zbl 0953.76069

Summary: We present a computational setting for the immersed boundary method employing an adaptive mesh refinement. Enhanced accuracy for the method is attained locally by covering an immersed boundary vicinity with a sequence of nested, progressively finer rectangular grid patches which dynamically follow the immersed boundary motion. The set of equations describing the interaction between a non-stationary, viscous incompressible fluid and an immersed elastic boundary is solved by coupling a projection method, specially designed for locally refined meshes, to an implicit formulation of the immersed boundary method.

The main contributions of this work concern the formulation and implementation of a multilevel self-adaptive version of the immersed boundary method on locally refined meshes. This approach is tested on a particular two-dimensional model problem, for which no significant difference is found between the solutions obtained on a mesh refined locally around the immersed boundary, and on the associated uniform mesh, built with the resolution of the finest level.

##### MSC:

76M20 | Finite difference methods (fluid mechanics) |

76D05 | Navier-Stokes equations (fluid dynamics) |

65M55 | Multigrid methods; domain decomposition (IVP of PDE) |

74F10 | Fluid-solid interactions |