In this paper, we introduce a \(C^{0}\) virtual element method for the Helmholtz transmission eigenvalue problem, which is a fourth-order non-selfadjoint eigenvalue problem. We consider the mixed formulation of the eigenvalue problem discretized by the lowest-order virtual elements. This discrete scheme is based on a conforming \(H^{1}(\varOmega )\times H^{1}(\varOmega )\) discrete formulation, which makes use of lower regular virtual element spaces. However, the discrete scheme is a non-classical mixed method due to the non-selfadjointness, then we cannot use the framework of classical eigenvalue problem directly. We employ the spectral theory of compact operator to prove the spectral approximation. Finally, some numerical results show that numerical eigenvalues obtained by the proposed numerical scheme can achieve the optimal convergence order.

在本文中，我们引入了针对Helmholtz传输特征值问题的\（C ^ {0} \）虚拟元素方法，该方法是四阶非自伴特征值问题。我们考虑由最低阶虚拟元素离散化的特征值问题的混合形式。此离散方案基于符合的\（H ^ {1}（\ varOmega）\乘以H ^ {1}（\ varOmega）\）离散公式，它使用较低的规则虚拟元素空间。但是，由于具有非自伴性，离散方案是一种非经典的混合方法，因此我们不能直接使用经典特征值问题的框架。我们采用紧算子的谱理论来证明谱近似。最后，一些数值结果表明，所提出的数值方案获得的数值特征值可以达到最优收敛阶。