A priori and a posteriori error analysis for a VEM discretization of the convection-diffusion eigenvalue problem

TL;DR

This paper introduces a virtual element method for a 2D convection-diffusion eigenvalue problem, providing error estimates and an adaptive refinement strategy tested on complex domains.

Contribution

It develops a novel VEM approach with proven a priori and a posteriori error estimates for non-symmetric eigenvalue problems.

Findings

Error estimates for eigenvalues and eigenfunctions

Reliable and efficient a posteriori error estimator

Effective adaptive refinement on irregular domains

Abstract

In this paper we propose and analyze a virtual element method for the two dimensional non-symmetric diffusion-convection eigenvalue problem in order to derive a priori and a posteriori error estimates. Under the classic assumptions of the meshes, and with the aid of the classic theory of compact operators, we prove error estimates for the eigenvalues and eigenfunctions. Also, we develop an a posteriori error estimator which, in one hand, results to be reliable and on the other, with standard bubble functions arguments, also results to be efficient. We test our method on domains where the complex eigenfunctions are not sufficiently regular, in order to assess the performance of the estimator that we compare with the uniform refinement given by the a priori analysis