Virtual Element Method for the Laplace-Beltrami equation on surfaces

Abstract

We present and analyze a Virtual Element Method (VEM) of arbitrary polynomial order $k\in \mathbb{N}$ for the Laplace-Beltrami equation on a surface in ${\mathbb{R}}^3$. The method combines the Surface Finite Element Method (SFEM) [Dziuk, Elliott, \emph{Finite element methods for surface PDEs}, 2013] and the recent VEM [Beirao da Veiga et al, \emph{Basic principles of Virtual Element Methods}, 2013] in order to handle arbitrary polygonal and/or nonconforming meshes. We account for the error arising from the geometry approximation and extend to surfaces the error estimates for the interpolation and projection in the virtual element function space. In the case $k=1$ of linear Virtual Elements, we prove an optimal $H^1$ error estimate for the numerical method. The presented method has the capability of handling the typically nonconforming meshes that arise when two ore more meshes are pasted along a straight line. Numerical experiments are provided to confirm the convergence result and to show an application of mesh pasting.

25 pages, 6 figures

References in corpus (1)

• Lumped finite element method for reaction-diffusion systems on compact surfaces

Cited by in corpus (6)

• Bulk-surface virtual element method for systems of PDEs in two-space dimension
• Weakly imposed Dirichlet boundary conditions for 2D and 3D Virtual Elements
• FETI-DP for the three-dimensional Virtual Element Method
• Virtual element method for elliptic bulk-surface PDEs in three space dimensions
• Matrix-oriented FEM formulation for stationary and time-dependent PDEs on x-normal domains
• A parametrix method for elliptic surface PDEs