Analysis of nonconforming virtual element method for the convection diffusion reaction equation with polynomial coefficients

TL;DR

This paper investigates the application of nonconforming virtual element methods to convection-diffusion-reaction equations with polynomial coefficients, establishing stability, well-posedness, and optimal convergence results.

Contribution

It extends the virtual element method analysis to non-symmetric cases using $L^2$ projection, proving well-posedness and polynomial consistency.

Findings

Proved stability for symmetric bilinear forms.

Established well-posedness for non-symmetric bilinear forms.

Achieved optimal convergence estimates in broken Sobolev norm.

Abstract

In this paper we discuss the application of nonconforming virtual element methods(VEM) for the second order diffusion dominated convection diffusion reaction equation. Stability of the virtual element methods has been proved for the symmetric bilinear form. But the same analysis cannot be carried out for the non-symmetric case. In this work we present the external virtual element methods using $L^2$ projection operator and prove the well-posedness of VEM for non symmetric bilinear form. We also proved polynomial consistency of discrete bilinear form assuming $H^2$ regularity of approximate solution on each triangle. We have shown optimal convergence estimate in the broken sobolev norm.