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

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 problems 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 derived 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.

An immersed finite element method for elliptic interface problems in three dimensions

“…For simplicity, we denote u s = u| Ω s , s = ±, in the rest of this article. The elliptic interface problem (1.1) has wide applications in science and engineering such as inverse problems [26,35,49], fluid dynamics [38,41], biomolecular electrostatics [21,53], plasma simulation [8,34], to name just a few. Traditional finite element methods can be applied to solve this interface problem based on an interface-fitted mesh [6,13,54].…”

“…In addition to the optimal convergence rate the IFE method can achieve on a unfitted mesh, it can also keep the number and location of degrees of freedom isomorphic to the standard finite element method defined on the same mesh. And this feature is advantageous when dealing with moving interface problems for which we refer readers to [3,7,26,33,42].…”

“…Although many works on the 2-D IFE methods have appeared in the literature, such as [22,23,25,27,28,32,43,44] for theoretical analysis and [2,3,7,26,51] for applications, to name just a few, the study on the 3-D case is relatively sparse, see [36] for a linear IFE method and [49] for a trilinear IFE method. Even though the research reported here is within the direction of that in [49], but our work has three distinct new contributions.…”

Three‐dimensional immersed finite‐element method for anisotropic magnetostatic/electrostatic interface problems with nonhomogeneous flux jump

“…The IFE‐PIC method has also been extended to handle unbounded interface problems with asymptotic boundary condition and periodic boundary condition . Furthermore, the IFE methods have been extended and applied to moving interface problems, interface inverse problems, and other interesting applications . However, the present IFE method can only be used to solve the equations with scalar coefficients, that is, isotropic interface problems.…”

Approximation capabilities of immersed finite element spaces for elasticity Interface problems

“…The key idea of an IFE space is to use standard polynomials on noninterface elements, but Hsieh-Clough-Tocher type [23,24] macro polynomials constructed according to interface jump conditions on interface elements. There have been quite a few publications on IFE methods, for example, IFE methods for elliptic interface problems are discussed in [25][26][27][28][29][30][31][32], IFE methods for interface problems of other types partial differential equations are presented in [33][34][35][36][37][38][39][40][41]. In particular, for planar-elasticity interface problems described by (1.2)-(1.6), a nonconforming linear IFE space on a uniform triangular mesh is discussed in [21,42,43].…”