Meng, Jian; Zhang, Yongchao; Mei, Liquan A virtual element method for the Laplacian eigenvalue problem in mixed form. (English) Zbl 1442.65388 Appl. Numer. Math. 156, 1-13 (2020). Summary: In this paper, the virtual element method for the approximation of Laplacian eigenvalue problem in mixed form is studied. We show that the discrete form satisfies the hypotheses required by the Brezzi-Babǔska theory. Under some assumptions on polygonal meshes, we employ the spectral theory of compact operators to prove the spectral approximation and the optimal order for the eigenvalues. Finally, some numerical results show that numerical eigenvalues obtained by the proposed numerical scheme can achieve the optimal convergence order. Cited in 10 Documents MSC: 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs 65N35 Spectral, collocation and related methods for boundary value problems involving PDEs 35P15 Estimates of eigenvalues in context of PDEs Keywords:virtual element method; polygonal meshes; Laplacian eigenvalue problem; spectral approximation Software:PolyMesher PDFBibTeX XMLCite \textit{J. Meng} et al., Appl. Numer. Math. 156, 1--13 (2020; Zbl 1442.65388) Full Text: DOI References: [1] An, J., A Legendre-Galerkin spectral approximation and estimation of the index of refraction for transmission eigenvalues, Appl. Numer. Math., 108, 171-184 (2016) · Zbl 1346.65057 [2] Antonietti, P. F.; Beirão Da Veiga, L.; Scacchi, S.; Verani, M., A \(C^1\) virtual element method for the Cahn-Hilliard equation with polygonal meshes, SIAM J. Numer. 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