Virtual Element Method for the Laplace-Beltrami equation on surfaces

Dipartimento di Matematica e Fisica E. De Giorgi, Università del Salento, via per Arnesano, 73100 Lecce, Italy

^* Corresponding author: massimo.frittelli@unisalento.it

Received: 2 December 2016
Accepted: 21 August 2017

Abstract

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.