Solution of Second Order Elliptic Equations on Polygonal/Polyhedral Meshes by the Virtual Element Method with Applications in Geosciences

The reaction-convection-diffusion equation is a second order elliptic partial differential equation that can be used to model many different physical phenomena. A method to solve this class of problems is the Virtual Element Method (VEM). In this thesis we focus on two dimensional problems defined on polygonal meshes, considering both the stationary and evolutive case. The aim of this thesis is to analyze and implement the stabilization methods Streamline Upwind Petrov-Galerkin (SUPG) and Mass Lumping in the particular case of Virtual Element space of order k = 1. Numerical results show the positive stabilization effect of SUPG and Mass Lumping when the problem is characterized respectively by very large Péclet and very low Karlowitz numbers. Moreover, an error analysis on an easy stationary problem shows that the stabilization methods preserve the rate of convergence of VEM. Finally, a simulation of a realistic geophysical evolutive problem is carried out to show the performance of the method on a domain characterized by a high geometrical complexity.