We present and analyze a Virtual Element Method (VEM) for the Laplace-Beltrami equation on a surface in ℝ3, that we call Surface Virtual Element Method (SVEM). The method combines the Surface Finite Element Method (SFEM) (Dziuk, Eliott, G. Dziuk and C.M. Elliott., Acta Numer. 22 (2013) 289–396.) and the recent VEM (Beirão da Veiga et al., Math. Mod. Methods Appl. Sci. 23 (2013) 199–214.) in order to allow for a general polygonal approximation of the surface. We account for the error arising from the geometry approximation and in the case of polynomial order k = 1 we extend to surfaces the error estimates for the interpolation in the virtual element space. We prove existence, uniqueness and first order H1 convergence of the numerical solution.We highlight the differences between SVEM and VEM from the implementation point of view. Moreover, we show that the capability of SVEM of handling nonconforming and discontinuous meshes can be exploited in the case of surface pasting. We provide some numerical experiments to confirm the convergence result and to show an application of mesh pasting.
Mots clés : Surface PDEs, Laplace-Beltrami equation, surface finite element method, Virtual Element Method
@article{M2AN_2018__52_3_965_0,
author = {Frittelli, Massimo and Sgura, Ivonne},
title = {Virtual {Element} {Method} for the {Laplace-Beltrami} equation on surfaces},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {965--993},
publisher = {EDP-Sciences},
volume = {52},
number = {3},
year = {2018},
doi = {10.1051/m2an/2017040},
mrnumber = {3865555},
zbl = {1456.65160},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2017040/}
}
TY - JOUR AU - Frittelli, Massimo AU - Sgura, Ivonne TI - Virtual Element Method for the Laplace-Beltrami equation on surfaces JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2018 SP - 965 EP - 993 VL - 52 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2017040/ DO - 10.1051/m2an/2017040 LA - en ID - M2AN_2018__52_3_965_0 ER -
%0 Journal Article %A Frittelli, Massimo %A Sgura, Ivonne %T Virtual Element Method for the Laplace-Beltrami equation on surfaces %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2018 %P 965-993 %V 52 %N 3 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2017040/ %R 10.1051/m2an/2017040 %G en %F M2AN_2018__52_3_965_0
Frittelli, Massimo; Sgura, Ivonne. Virtual Element Method for the Laplace-Beltrami equation on surfaces. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 3, pp. 965-993. doi : 10.1051/m2an/2017040. http://www.numdam.org/articles/10.1051/m2an/2017040/
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