A lowest-order virtual element method for the Helmholtz transmission eigenvalue problem

In this paper, we introduce a C0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^{0}$$\end{document} 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 H1(Ω)×H1(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^{1}(\varOmega )\times H^{1}(\varOmega )$$\end{document} 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.