Virtual Element Method for the Laplace-Beltrami equation on surfaces
This work extends VEM to surface PDEs, enabling the use of nonconforming meshes for problems on surfaces, which is useful for computational geometry and engineering applications.
The authors present a Virtual Element Method (VEM) for solving the Laplace-Beltrami equation on surfaces, achieving optimal H1 error estimates for linear elements and handling arbitrary polygonal and nonconforming meshes. Numerical experiments confirm convergence and demonstrate mesh pasting applications.
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.