In this paper, we use the non-conforming Crouzeix-Raviart element method to solve a Stekloff eigenvalue problem arising in inverse scattering. The weak formulation corresponding to this problem is non-self-adjoint and indefinite, and its Crouzeix-Raviart element discretization does not meet the condition of the Strang lemma. We use the standard duality technique to prove an extension of the Strang lemma. And we prove the convergence and error estimate of discrete eigenvalues and eigenfunctions using the spectral perturbation theory for compact operators. Finally, we present some numerical examples not only on uniform meshes but also on adaptive refined meshes to show that the Crouzeix-Raviart method is efficient for computing real and complex eigenvalues as expected.

在本文中，我们使用非协调Crouzeix-Raviart元素方法来解决逆散射中产生的Stekloff特征值问题。与此问题相对应的弱公式是非自伴且不确定的，并且其Crouzeix-Raviart元素离散化不满足Strang引理的条件。我们使用标准对偶技术来证明Strang引理的扩展。并利用谱摄动理论为紧算子证明了离散特征值和特征函数的收敛性和误差估计。最后，我们不仅在均匀网格上而且还在自适应细化网格上提供一些数值示例，以表明Crouzeix-Raviart方法可有效地按预期方式计算实际和复杂特征值。