In this paper, we propose a conforming virtual element method for the Helmholtz transmission eigenvalue problem of anisotropic media. By using $𝕋$-coercivity theory, the spectral approximation theory of compact operator and the projection and interpolation error estimates, we prove the spectral convergence of the discrete scheme and the optimal a priori error estimates for the discrete eigenvalues and eigenfunctions. The virtual element method has great flexibility in handling polygonal meshes, which motivates us to construct a fully computable a posteriori error estimator for the virtual element method. Then the upper bound of the approximation error is derived from the residual equation and the inf-sup condition. In turn, the related lower bound is established by using the bubble function strategy. Finally, we provide numerical examples to verify the theoretical results, including the optimal convergence of the virtual element scheme on uniformly refined meshes and the efficiency of the estimator on adaptively refined meshes.

在本文中，我们针对各向异性介质的亥姆霍兹传输特征值问题提出了一种一致的虚拟元方法。通过使用$𝕋$-矫顽力理论，紧算子的谱逼近理论和投影和插值误差估计，我们证明了离散格式的谱收敛性和离散特征值和特征函数的最优先验误差估计。虚拟元方法在处理多边形网格方面具有很大的灵活性，这促使我们构建一个完全可计算的后验虚元法的误差估计器。然后从残差方程和 inf-sup 条件推导出逼近误差的上界。反过来，通过使用气泡函数策略建立相关的下界。最后，我们提供数值例子来验证理论结果，包括虚拟单元方案在均匀细化网格上的最佳收敛性和估计器在自适应细化网格上的效率。