Speaker
Description
The Virtual Element Method (VEM) is a polygonal finite element method characterized by geometric flexibility and has therefore been applied to a wide range of engineering problems. Despite its popularity, some difficulties arise when dealing with strongly nonlinear and anisotropic problems, mainly due to the presence of a non-consistent stabilization term that needs to be introduced to ensure the well-posedness of the discrete problem.
Starting from the work in [1], in this talk we present the higher-order Stabilization-Free Virtual Element Method for general second order elliptic problems [2]. The proposed approach is based on a new polynomial projection that enables the construction of an operator-preserving scheme. Moreover, this new discretization allows the derivation of stabilization-free a posteriori error estimates, a result that cannot be achieved for the classical VEM on general polygonal meshes. The proposed framework opens new perspectives and applications also for higher-order problems, such as the biharmonic equation.
References
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BERRONE, Stefano; BORIO, Andrea; MARCON, Francesca. A stabilization-free virtual element method based on divergence-free projections. Comput. Methods Appl. Mech. Engrg. 2024, vol. 424, Paper No. 116885, 19. issn 0045-7825, issn 1879-2138. Available from doi: 10.1016/j.cma.2024. 116885.
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BERRONE, Stefano; BORIO, Andrea; FASSINO, Davide; MARCON, Francesca. Stabilization-free virtual element method for 2D second order elliptic equations. Comput. Methods Appl. Mech. Engrg. 2025, vol. 438, Paper No. 117839, 24. issn 0045-7825, issn 1879-2138. Available from doi: 10.1016/j.cma.2025.117839.