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Description
Virtual Elements (VEM) [1] generalize the Finite Element Method, allowing the discretization of complex geometries through general polytopal meshes.
The discrete space, which includes polynomials to ensure accuracy, is implicitly defined, as shape functions are solutions of a local PDE that is typically not solved. The discrete problem is constructed via polynomial projections, and well-posedness is ensured by a stabilization term accounting for the non-polynomial part of the space. However, the stabilization form is generally arbitrary and may not reflect the physical properties of the problem. Consequently, stabilization-free [2] or self-stabilized [3] VEM formulations are gaining popularity.
Moreover, an improper choice of polynomial projector and stabilization may deteriorate accuracy in the presence of variable coefficients.
This study [4] investigates the p-version of VEM and compares stabilized and self-stabilized formulations on academic benchmarks and application-oriented scenarios, particularly curvilinearly stiffened variable stiffness panels, widely used in aerospace structures.
References
[1] L. Beirão da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini, A. Russo. Basic principles of virtual element methods. Mathematical Models and Methods in Applied Sciences, 23(01), 199-214, 2013.
[2] S. Berrone, A. Borio, F. Marcon. Lowest order stabilization free virtual element method for the 2D Poisson equation. Computers & Mathematics with Applications, 177, 78-99 , 2025.
[3] A. Lamperti, M. Cremonesi, U. Perego, A. Russo, C. Lovadina. A Hu–Washizu variational approach to self-stabilized virtual elements: 2D linear elastostatics. Computational Mechanics, 71(5), 935-955 , 2023.
[4] P. P. Foligno, D. Boffi, F. Credali, R. Vescovini. Benchmarking stabilized and self-stabilized p-virtual element methods with variable coefficients arXiv preprint, arXiv:2511.18943, 2025. Computer Methods in Applied Mechanics and Engineering, to appear.