A Nonconforming Virtual Element Method for Advection-Diffusion-Reaction Problems with CIP Stabilization

TL;DR

This paper introduces a nonconforming virtual element method with CIP stabilization for advection-diffusion-reaction problems, providing stability analysis and error estimates without skew-symmetrization.

Contribution

It presents a novel VEM approach with CIP stabilization and Nitsche boundary conditions, along with stability proof and error estimates for these complex PDEs.

Findings

Proves stability of the proposed VEM method.

Derives $h$-version error estimates.

Demonstrates effectiveness for advection-diffusion-reaction problems.

Abstract

We study a nonconforming virtual element method (VEM) for advection-diffusion-reaction problems with continuous interior penalty (CIP) stabilization. The design of the method is based on a standard variational formulation of the problem (no skew-symmetrization), and boundary conditions are imposed with a Nitsche technique. We use the enhanced version of VEM, with a ``DoFi-DoFi'' stabilization in the diffusion and reaction terms. We prove stability of the proposed method and derive $h$-version error estimates.