VEM discretization allowing small edges for the reaction-convection-diffusion equation: source and spectral problems

TL;DR

This paper introduces a virtual element method that accommodates small edges in polygonal meshes for reaction-convection-diffusion problems, providing stability, error estimates, and numerical validation.

Contribution

It extends virtual element methods to meshes with small edges, offering new stability and error analysis for both load and spectral problems.

Findings

Method is stable and convergent with small edges

Error estimates are derived for eigenvalues and eigenfunctions

Numerical tests confirm theoretical results

Abstract

In this paper we analyze a lowest order virtual element method for the load classic reaction-convection-diffusion problem and the convection-diffusion spectral problem, where the assumptions on the polygonal meshes allow to consider small edges for the polygons. Under well defined seminorms depending on a suitable stabilization for this geometrical approach, we derive the well posedness of the numerical scheme and error estimates for the load problem, whereas for the spectral problem we derive convergence and error estimates for the eigenvalues and eigenfunctions. We report numerical tests to assess the performance of the small edges on our numerical method for both problems under consideration.