In this paper, we propose and analyze the least squares finite element methods for the linear elasticity interface problem in the stress-displacement system on unfitted meshes. We consider the cases that the interface is $C^2$ or polygonal, and the exact solution $(\sigma,u)$ belongs to $H^s(div; \Omega_0 \cup \Omega_1) \times $H^{1+s}(\Omega_0 \cup \Omega_1)$ with $s > 1/2$. Two types of least squares functionals are defined to seek the numerical solution. The first is defined by simply applying the $L^2$ norm least squares principle, and requires the condition $s \geq 1$. The second is defined with a discrete minus norm, which is related to the inner product in $H^{-1/2}(\Gamma)$. The use of this discrete minus norm results in a method of optimal convergence rates and allows the exact solution has the regularity of any $s > 1/2$. The stability near the interface for both methods is guaranteed by the ghost penalty bilinear forms and we can derive the robust condition number estimates. The convergence rates under $L^2$ norm and the energy norm are derived for both methods. We illustrate the accuracy and the robustness of the proposed methods by a series of numerical experiments for test problems in two and three dimensions.

翻译：本文针对非拟合网格上应力-位移系统的线性弹性界面问题，提出并分析最小二乘有限元方法。我们考虑界面为$C^2$光滑或多边形的情况，且精确解$(\sigma,u)$属于$H^s(div; \Omega_0 \cup \Omega_1) \times H^{1+s}(\Omega_0 \cup \Omega_1)$（其中$s > 1/2$）。定义两类最小二乘泛函以求解数值解：第一类通过直接应用$L^2$范数最小二乘原理定义，要求条件$s \geq 1$；第二类采用离散负范数定义（与$H^{-1/2}(\Gamma)$内积相关），该离散负范数的使用使方法达到最优收敛速率，并允许精确解具有任意$s > 1/2$的正则性。两种方法在界面附近的稳定性由鬼罚双线性形式保证，可推导出鲁棒的条件数估计。同时推导了两种方法在$L^2$范数和能量范数下的收敛速率。通过二维和三维测试问题的系列数值实验，验证了所提方法的精度与鲁棒性。