A first-order stabilization-free virtual element method. (English) Zbl 07708812
Summary: In this paper, we introduce a new Virtual Element Method (VEM) not requiring any stabilization term based on the usual enhanced first-order VEM space. The new method relies on a modified formulation of the discrete diffusion operator that ensures stability preserving all the properties of the differential operator.
MSC:
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
| 65N15 | Error bounds for boundary value problems involving PDEs |
| 65N50 | Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs |
| 52C20 | Tilings in \(2\) dimensions (aspects of discrete geometry) |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
References:
| [1] | Berrone, S.; Borio, A.; Marcon, F., Lowest order stabilization free virtual element method for the 2D Poisson equation (2021), arXiv:2103.16896 |
| [2] | Berrone, S.; Borio, A.; Marcon, F., Comparison of standard and stabilization free virtual elements on anisotropic elliptic problems, App. Math. Lett., 129, Article 107971 pp. (2022) · Zbl 1487.65176 |
| [3] | Beirão da Veiga, L.; Brezzi, F.; Marini, L. D.; Russo, A., Virtual element methods for general second order elliptic problems on polygonal meshes, Math. Models Methods Appl. Sci., 26, 04, 729-750 (2015) · Zbl 1332.65162 |
| [4] | Perot, J. B.; Chartrand, C., A mimetic method for polygons, J. Comput. Phys., 424, C, Article 109853 pp. (2021) · Zbl 07508458 |
| [5] | Beirão da Veiga, L.; Manzini, G., Residual a posteriori error estimation for the virtual element method for elliptic problems, ESAIM: M2AN, 49, 2, 577-599 (2015) · Zbl 1346.65056 |
| [6] | Berrone, S., Robustness in a posteriori error estimates for the oseen equations with general boundary conditions, (Numerical Mathematics and Advanced Applications (2003), Springer Milan), 657-668 · Zbl 1254.76097 |
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