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

We present a new fixed mesh method for solving a class of interface inverse problems for the typical elliptic interface problems. These interface inverse problems are formulated as shape optimization problems. By an immersed finite element (IFE) method, both the governing partial differential equations and the objective functional for an interface inverse problem are discretized optimally regardless of the location of the interface in a chosen mesh, and the shape optimization for recovering the interface is reduced to a constrained optimization problem. The formula for the gradient of the objective function in this constrained optimization is derived and this formula can be implemented efficiently in the IFE framework. As demonstrated by three representative applications, the proposed IFE method can be employed to solve a spectrum of interface inverse problems efficiently.