An overview of the immersed interface method and its applications.

*(English)*Zbl 1028.65108Summary: Interface problems have many applications. Mathematically, interface problems usually lead to differential equations whose input data and solutions are non-smooth or discontinuous across some interfaces. The immersed interface method (IIM) has been developed in recent years particularly designed for interface problems. The IIM is a sharp interface method based on Cartesian grids. The IIM makes use of the jump conditions across the interface so that the finite difference/element discretization can be accurate.

In this survey paper, we will introduce the immersed interface method for various problems, discuss its recent advances and related software packages, and some of its applications. We also review some other related methods and references in this survey paper.

In this survey paper, we will introduce the immersed interface method for various problems, discuss its recent advances and related software packages, and some of its applications. We also review some other related methods and references in this survey paper.

##### MSC:

65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |

35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |

35K05 | Heat equation |

35J25 | Boundary value problems for second-order elliptic equations |

35K55 | Nonlinear parabolic equations |

35R05 | PDEs with low regular coefficients and/or low regular data |

35Q30 | Navier-Stokes equations |

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

65N06 | Finite difference methods for boundary value problems involving PDEs |

65M55 | Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs |

65Y15 | Packaged methods for numerical algorithms |