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

Carlo Lovadina, Ilaria Perugia, Manuel Trezzi

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.

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

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 -version error estimates.
Paper Structure (13 sections, 19 theorems, 154 equations, 4 figures)

This paper contains 13 sections, 19 theorems, 154 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The Continuous and the Discrete Problem
  3. Continuous Problem
  4. Notation and Polynomial Approximation Results
  5. Virtual Element Spaces
  6. Virtual Element Forms and the Discrete Problem
  7. Theoretical Analysis
  8. Preliminary results
  9. Inf-Sup condition
  10. Error Estimates
  11. Numerical results
  12. Convergence rates of the schemes.
  13. Robustness with respect to the parameter $\varepsilon$.

Key Result

Lemma 2.1

Under assumption (A1), for any $E \in \Omega_h$ and for any sufficiently smooth function $\varphi$ defined on $E$, we have that

Figures (4)

  • Figure 1: Degrees of freedom for a penthagon.
  • Figure 2: Example of meshes used for the tests.
  • Figure 3: Convergences for $u_1$ (left column) and $u_2$(right column).
  • Figure 4: Results for various choice of $\varepsilon$.

Theorems & Definitions (38)

  • Lemma 2.1: Polynomial approximation
  • Lemma 2.2
  • proof
  • Remark 2.1
  • Lemma 2.3: Approximation with nonconforming virtual element functions
  • Proposition 3.1: Inverse inequality
  • proof
  • Lemma 3.1: Inverse trace inequality
  • proof
  • Theorem 3.1
  • ...and 28 more