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

TL;DR

This paper introduces a fixed mesh immersed finite element method for efficiently solving various interface inverse problems, enabling accurate shape optimization without remeshing.

Contribution

The paper develops a novel fixed mesh IFE approach with explicit gradient formulas for interface inverse problems, improving computational efficiency and accuracy.

Findings

Successfully applied to three different interface inverse problems.

Achieved optimal discretization of PDEs and objective functionals.

Provided explicit gradient formulas for shape optimization.

Abstract

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