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

Abstract

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.

Appendix A Technical Results

1.1 A.1 Proof of Lemma 3.1

First, differentiating $x_P=x(\hat{t}_P({\varvec{\alpha }}),{\varvec{\alpha }})$ and $y_P=y(\hat{t}_P({\varvec{\alpha }}),{\varvec{\alpha }})$ with respect to $\alpha _j$ and letting that $\frac{\partial x}{\partial \hat{t}_P} := \frac{\partial x}{\partial t}|_{t=\hat{t}_P}$ and $\frac{\partial y}{\partial \hat{t}_P} := \frac{\partial y}{\partial t}|_{t=\hat{t}_P}$, we have $D_{\alpha _j}x_P= \frac{\partial x}{\partial \hat{t}_P} \frac{\partial \hat{t}_P}{ \partial \alpha _j } +\frac{\partial x}{\partial \alpha _j}|_{t=\hat{t}_P},~ D_{\alpha _j}y_P= \frac{\partial y}{\partial \hat{t}_P} \frac{\partial \hat{t}_P}{ \partial \alpha _j } +\frac{\partial y}{\partial \alpha _j}|_{t=\hat{t}_P},$ which leads to

$$\begin{aligned} \frac{\partial y}{\partial \hat{t}_P} D_{\alpha _j}x_P - \frac{\partial x}{\partial \hat{t}_P} D_{\alpha _j}y_P = \frac{\partial y}{\partial \hat{t}_P}\frac{\partial x}{\partial \alpha _j}|_{t=\hat{t}_P} - \frac{\partial x}{\partial \hat{t}_P}\frac{\partial y}{\partial \alpha _j}|_{t=\hat{t}_P}. \end{aligned}$$

(A.1)

On the other hand, since P is on the edge $A_1A_2$, we have the equation $(y_2-y_1)x_P-(x_2-x_1)y_P=x_2y_1-x_1y_2.$ Differentiating it with respect to $\alpha _j$ yields

$$\begin{aligned} (y_2-y_1)D_{\alpha _j}x_P -(x_2-x_1)D_{\alpha _j}y_P=0. \end{aligned}$$

(A.2)

Combining (A.2) and (A.1) yields the linear system for $D_{\alpha _j}P$ in (3.8). Let $\mathbf { n}_e$ be the normal vector to the edge $A_1A_2$. Then we have $\text {det}(M_P(\hat{t}_P))=\mathbf { n}_e\cdot \nabla \Gamma (\hat{t}_P({\varvec{\alpha }}), {\varvec{\alpha }})$ which is non zero by the assumption that $A_1A_2$ is not tangent to $\Gamma (t, {\varvec{\alpha }})$ at P.

1.2 A.2 Material Derivatives of Local Matrices and Vectors

For the simplicity, we assume that the boundary condition functions $g^k_N$, $g^k_D$ and the force term $f^k$ are fixed and independent with interface change, $1\leqslant k \leqslant K$. Therefore, on each interface element $T\in \mathcal {T}_h^{int}$ and each interface edge $e \in \mathcal {E}_h^{int}$, we have:

$$\begin{aligned} D_{\alpha _j}\mathbf { K}_T&= \left( \int _T \beta \nabla \frac{\partial \psi _{p,T}}{\partial \alpha _j} \cdot \nabla \psi _{q,T} dX \right) _{p, q\in \mathcal {I}} + \left( \int _T \beta \nabla \frac{\partial \psi _{p,T}}{\partial \alpha _j} \cdot \nabla \psi _{q,T} dX \right) _{p, q\in \mathcal {I}}^T \nonumber \\&\quad +\left( \sum _{m=1}^3 \int _{T_m} \beta \nabla \psi _{p,T} \cdot \nabla \psi _{q,T}~ dX ~ tr \left( (D_{\alpha _j} \mathbf { J}_m) \mathbf { J}^{-1}_m \right) \right) _{p, q\in \mathcal {I}}, \end{aligned}$$

(A.3a)

$$\begin{aligned} D_{\alpha _j} \mathbf { E}^{r_1r_2}_e&= \left( \int _e \beta \nabla \frac{\partial \psi _{p,T^{r_1}}}{\partial \alpha _j}\cdot (\psi _{q,T^{r_2}}\mathbf { n}^{r_2}_e) ds \right) _{p, q\in \mathcal {I}} \nonumber \\&\quad + \left( \int _e \beta \nabla \psi _{p,T^{r_1}}\cdot \left( \frac{\partial \psi _{q,T^{r_2}}}{\partial \alpha _j}\mathbf { n}^{r_2}_e\right) ds \right) _{p, q\in \mathcal {I}} \nonumber \\&\quad + \Bigg ( \beta ^- \nabla \psi ^-_{p,T^{r_1}}\cdot (\psi ^-_{q,T^{r_2}}\mathbf { n}^{r_2}_e)|_{P} - \beta ^+ \nabla \psi ^+_{p,T^{r_1}}\cdot (\psi ^+_{q,T^{r_2}}\mathbf { n}^{r_2}_e)|_{P} \Bigg )_{p, q\in \mathcal {I}} \nonumber \\&\quad \frac{ D_{\alpha _j}P\cdot (A_2-A_1)}{\left\| A_2-A_1\right\| }, \end{aligned}$$

(A.3b)

$$\begin{aligned} D_{\alpha _j} \mathbf { G}^{r_1r_2}_e&= \frac{\sigma ^0_e}{|e|} \left( \int _e \left( \frac{\partial \psi _{p,T^{r_1}}}{\partial \alpha _j}\mathbf { n}^{r_1}_e\right) \cdot (\psi _{q,T^{r_2}}\mathbf { n}^{r_2}_e) ds \right. \nonumber \\&\quad \left. + \int _e (\psi _{p,T^{r_1}}\mathbf { n}^{r_1}_e)\cdot \left( \frac{\partial \psi _{q,T^{r_2}}}{\partial \alpha _j}\mathbf { n}^{r_2}_e\right) ds \right) _{p, q\in \mathcal {I}}, \end{aligned}$$

(A.3c)

$$\begin{aligned} D_{\alpha _j} \mathbf {R}_T&= \left( \int _T \frac{\partial \psi _{p,T}}{\partial \alpha _j} dX + \int _T\nabla \psi _{p,T}\cdot \mathbf { V}^j dX \right) _{p\in \mathcal {I}} \nonumber \\&\quad +\left( \sum _{i=1}^3 \int _{T_i} \psi _{p,T}~ dX ~ tr \left( (D_{\alpha _j} \mathbf { J}_i) \mathbf { J}^{-1}_i \right) \right) _{p \in \mathcal {I}}. \end{aligned}$$

(A.3d)

and the material derivatives of vectors:

$$\begin{aligned} D_{\alpha _j}\mathbf { F}_T^k&= \left( \int _T f^k \frac{\partial \psi _{p,T}}{\partial \alpha _j} dX \right) _{p\in \mathcal {I}} + \left( \int _T \nabla (f^k\psi _{p,T})\cdot \mathbf { V}^j_T dX \right) _{p\in \mathcal {I}} \nonumber \\&\quad +\left( \sum _{m=1}^3 \int _{T_m} f^k \psi _{p,T}~ dX ~ tr \left( (D_{\alpha _j} \mathbf { J}_m) \mathbf { J}^{-1}_m \right) \right) _{p \in \mathcal {I}}, \end{aligned}$$

(A.4a)

$$\begin{aligned} D_{\partial _j} \mathbf { B}_e^k&= \left( \int _e \beta g_D^k \nabla \frac{\partial \psi _{p,T}}{\partial \alpha _j}\cdot \mathbf { n}_e ds \right) _{p\in \mathcal {I}}\nonumber \\&\quad + \Bigg ( \beta ^- g_D^k \nabla \psi ^-_{p,T}\cdot \mathbf { n}_e|_{P} - \beta ^+ g_D^k \nabla \psi ^+_{p,T}\cdot \mathbf { n}_e|_{P} \Bigg )_{p\in \mathcal {I}} \frac{ D_{\alpha _j}P\cdot (A_2-A_1)}{\left\| A_2-A_1\right\| }, \end{aligned}$$

(A.4b)

$$\begin{aligned} D_{\partial _j} \mathbf { C}_e^k&= \frac{\sigma ^0_e}{|e|}\left( \int _e \beta g_D^k \frac{\partial \psi _{p,T}}{\partial \alpha _j} ds \right) _{p\in \mathcal {I}} + \frac{\sigma ^0_e}{|e|}\Bigg ( \beta ^- g_D^k \psi ^-_{p,T}|_{P} - \beta ^+ g_D^k \psi ^+_{p,T}|_{P} \Bigg )_{p\in \mathcal {I}} \nonumber \\&\quad \frac{ D_{\alpha _j}P\cdot (A_2-A_1)}{\left\| A_2-A_1\right\| }, \end{aligned}$$

(A.4c)

$$\begin{aligned} D_{\partial _j} \mathbf { N}_e^k&= \left( \int _e g_N^k \frac{\partial \psi _{p,T}}{\partial \alpha _j} ds \right) _{p\in \mathcal {I}} + \Bigg ( g_N^k \psi ^-_{p,T}|_{P} - g_N^k \psi ^+_{p,T}|_{P} \Bigg )_{p\in \mathcal {I}} \frac{ D_{\alpha _j}P\cdot (A_2-A_1)}{\left\| A_2-A_1\right\| }. \end{aligned}$$

(A.4d)