Massimo Frittelli

Massimo Frittelli, Ivonne Sgura

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.

Massimo Frittelli, Anotida Madzvamuse, Ivonne Sgura et al.

We propose and analyse a novel surface finite element method that preserves the invariant regions of systems of semilinear parabolic equations on closed compact surfaces in $\mathbb{R}^3$ under discretisation. We also provide a fully-discrete scheme by applying the implicit-explicit (IMEX) Euler method in time. We prove the preservation of the invariant rectangles of the continuous problem under spatial and full discretizations. For scalar equations, these results reduce to the well-known discrete maximum principle. Furthermore, we prove optimal error bounds for the semi- and fully-discrete methods, that is the convergence rates are quadratic in the meshsize and linear in the timestep. Numerical experiments are provided to support the theoretical findings. In particular we provide examples in which, in the absence of lumping, the numerical solution violates the invariant region leading to blow-up due to the nature of the kinetics.