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In many applications, we often need to simulate a procedure in a domain consisting of several materials separated from each other by curves or surfaces. This usually leads to a so called interface problem involving partial differential equations whose coefficients are discontinuous across the material interface. Conventional finite element methods can be used to solve interface problems, but they usually require that each element contains essentially one of the materials.
Geometrically, this means that each element has to be on one side of a material interface. The recently developed immersed finite element (IFE) methods allow each element to contain multiple materials such that elements can sit on an interface.
The basic features of IFE methods are: (A) Their meshes can be independent of the interface location; hence, if preferred, structured meshes can be used to solve a problem with non-trivial interface. (B) The shape functions on each non-interface element are the same polynomials as those used by traditional finite element methods. (C) The shape functions on each interface element are macro polynomials constructed according the interface jump conditions. IFE methods can efficiently solve interface problems, especially those with a moving interface.
As a prototype application, IFE methods have been employed in the shape optimization framework to solve some interface inverse problems.