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We present a new fixed mesh algorithm for solving a class of interfaceinverse problems for the typical elliptic interface problems. These interfaceinverse problems are formulated as shape optimization prob- lems whoseobjective functionals depend on the shape of the interface. Regardless of thelocation of the interface, both the governing partial differential equationsand the objective functional are discretized optimally, with respect to theinvolved polynomial space, by an immersed finite element (IFE) method on afixed mesh. Furthermore, the formula for the gradient of the descritizedobjective function is de- rived within the IFE framework that can be computedaccurately and efficiently through the discretized adjoint procedure. Featuresof this proposed IFE method based on a fixed mesh are demonstrated by itsapplications to three representative interface inverse problems: the interfaceinverse problem with an internal measurement on a sub-domain, aDirichlet-Neumann type inverse problem whose data is given on the boundary, anda heat dissipation design problem.

A Fixed Mesh Method With Immersed Finite Elements for Solving Interface Inverse Problems