Virtual element method
The Virtual Element Method (VEM), proposed in 2013,[1] is a numerical technique used for solving partial differential equations (PDEs).[2][3] It is a generalization of the Finite Element Method (FEM) and is particularly noted for its flexibility in handling complex geometries.
VEM allows the use of general polygonal and polyhedral meshes, accommodating elements with any number of sides.[4][3] This flexibility simplifies the meshing process for intricate geometries. The method draws inspiration from Mimetic Finite Difference schemes, which aim to replicate the properties of differential operators at the discrete level. Additionally, VEM supports high polynomial degrees, enhancing the accuracy of the solutions.
Principles
The Virtual Element Method (VEM) is a technique for numerically approximating partial differential equations using generalised polygonal or polytopedral meshes. It is an extension of Mimetic Finite Differences (MFD) and Finite Element Methods (FEM).[1]
Unlike traditional finite element methods that require convex element shapes, VEM can handle polygonal and polyhedral elements of arbitrary shape, including non-convex elements and elements with holes. VEM ensures that polynomials up to a certain degree are reproduced exactly, providing accuracy guarantees similar to finite element methods. The method incorporates stabilization terms to ensure numerical stability without compromising accuracy. Central to VEM is the use of projection operators that map functions from the virtual element space onto polynomial spaces.
With a proper choice of discretization, VEM can ensure any desired degree of continuity. This, combined with the arbitrarily shapes polygonal and polyhedral elements, constitutes a significant advantage over FEM.
References
- ↑ 1.0 1.1 BeirãO Da Veiga, L.; Brezzi, F.; Cangiani, A.; Manzini, G.; Marini, L. D.; Russo, A. (January 2013). "Basic Principles of Virtual Element Methods". Mathematical Models and Methods in Applied Sciences. 23 (1): 199–214. doi:10.1142/S0218202512500492. ISSN 0218-2025.
- ↑ Veiga, Lourenço Beirão Da; Brezzi, Franco; Marini, L. Donatella; Russo, Alessandro (May 2023). "The virtual element method". Acta Numerica. 32: 123–202. doi:10.1017/S0962492922000095. ISSN 0962-4929.
- ↑ 3.0 3.1 Antonietti, Paola F.; Beirão da Veiga, Lourenço; Manzini, Gianmarco, eds. (2022). "The Virtual Element Method and its Applications". SEMA SIMAI Springer Series. 31. doi:10.1007/978-3-030-95319-5. ISBN 978-3-030-95318-8. ISSN 2199-3041.
- ↑ Beirão da Veiga, L.; Brezzi, F.; Marini, L. D.; Russo, A. (2014). "The Hitchhiker's Guide to the Virtual Element Method". Mathematical Models and Methods in Applied Sciences. 24 (8): 1541–1573. doi:10.1142/S021820251440003X. ISSN 0218-2025.
Further reading
- Dassi, F. (2023). "Vem++, a c++ library to handle and play with the Virtual Element Method". arXiv:2310.05748 [math.NA].
- Beirão Da Veiga, Lourenço; Brezzi, Franco; Marini, L. Donatella; Russo, Alessandro (2023). "The virtual element method". Acta Numerica. 32: 123–202. doi:10.1017/S0962492922000095.
- The Virtual Element Method and its Applications. SEMA SIMAI Springer Series. 31. 2022. doi:10.1007/978-3-030-95319-5. ISBN 978-3-030-95318-8. Search this book on
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