Virtual element method for the Laplace-Beltrami equation on surfaces

We present and analyze a Virtual Element Method (VEM) for the Laplace-Beltrami equa- tion on a surface in R3, that we call Surface Virtual Element Method (SVEM). The method combines the Surface Finite Element Method (SFEM) [Dziuk, Eliott, Finite element methods for surface PDEs, 2013] and the recent VEM [Beirao da Veiga et al, Basic principles of virtual element methods, 2013] in order to allow 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.