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A three-level CIP-VEM approach for the Oseen equation

Manuel Trezzi

TL;DR

The paper introduces a pressure-robust virtual element method for the Oseen problem stabilized by a three-level CIP mechanism to remain stable in advection-dominated regimes. By exploiting a divergence-free VE framework, an Oswald-based analysis in a stream-function space, and careful inverse/trace estimates, the authors establish stability and derive $h$-version error bounds for velocity and pressure. Numerical experiments on polygonal/Voronoi meshes confirm optimal convergence rates, demonstrate pressure-robustness, and illustrate the CIP stabilization’s effectiveness in boundary-layer and pipe-flow–like scenarios. The method shows promise for complex geometries and advection-dominated flows common in incompressible fluid dynamics, with a rigorous theoretical foundation and corroborating computational results.

Abstract

We study a pressure-robust virtual element method for the Oseen problem. In the advection-dominated case, the method is stabilized with a three level jump of the convective term. To analyze the method, we prove specific estimates for the virtual space of potentials. Finally, e prove stability of the proposed method in the advection-dominated limit and derive h-version error estimates for the velocity and the pressure.

A three-level CIP-VEM approach for the Oseen equation

TL;DR

The paper introduces a pressure-robust virtual element method for the Oseen problem stabilized by a three-level CIP mechanism to remain stable in advection-dominated regimes. By exploiting a divergence-free VE framework, an Oswald-based analysis in a stream-function space, and careful inverse/trace estimates, the authors establish stability and derive -version error bounds for velocity and pressure. Numerical experiments on polygonal/Voronoi meshes confirm optimal convergence rates, demonstrate pressure-robustness, and illustrate the CIP stabilization’s effectiveness in boundary-layer and pipe-flow–like scenarios. The method shows promise for complex geometries and advection-dominated flows common in incompressible fluid dynamics, with a rigorous theoretical foundation and corroborating computational results.

Abstract

We study a pressure-robust virtual element method for the Oseen problem. In the advection-dominated case, the method is stabilized with a three level jump of the convective term. To analyze the method, we prove specific estimates for the virtual space of potentials. Finally, e prove stability of the proposed method in the advection-dominated limit and derive h-version error estimates for the velocity and the pressure.

Paper Structure

This paper contains 16 sections, 27 theorems, 161 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Model problem
  3. The method
  4. Mesh assumptions
  5. Virtual element spaces
  6. Virtual element forms and the discrete problem
  7. Theoretical analysis
  8. Preliminary results
  9. Coercivity
  10. Error analysis
  11. Pressure error analysis
  12. Numerical results
  13. Solution with a boundary layer
  14. Convergence analysis.
  15. Pressure robustness.
  16. ...and 1 more sections

Key Result

Lemma 3.1

Let $\mathbf{v} \in \mathbf{Z}(\Omega) \cap [H^s(\Omega_h)]^2$ be a smooth function with $s \geq 1$. There exists a potential $\psi \in \Phi(\Omega) \cap H^{s+1}(\Omega_h)$ such that $\textnormal{curl}(\psi) = \mathbf{v}$ and

Figures (5)

  • Figure 1: Numerical solutions obtained with different choices of the triple $(\delta_1, \delta_2, \delta_3)$.Top-left $(\delta_1, \delta_2, \delta_3) = (0,0,0)$, top-right $(\delta_1, \delta_2, \delta_3) = (0.1,0,0)$, bottom-left $(\delta_1, \delta_2, \delta_3) = (0.1,0.01,0)$, bottom-right $(\delta_1, \delta_2, \delta_3) = (0.1,0.01,0.01)$ .
  • Figure 2: Convergences results for the velocity $\mathbf{u}$ in the $L^2$ norm (left column) and in the $H^1$-seminorm (right column). The red lines correspond to the case $k=2$, the blue lines to the case $k=3$, and the green lines to the case $k=4$.
  • Figure 3: Convergences for the pressure $p$ in the $L^2$-norm. The red lines correspond to the case $k=2$, the blue lines to the case $k=3$, and the green lines to the case $k=4$.
  • Figure 4: Result for the $L^2-$norm of the error (left column) and the $H^1-$seminorm of the error (right column). The red lines correspond to the case $k=2$, the blue lines to the case $k=3$, and the green lines to the case $k=4$.
  • Figure 5: Numerical representation of a fluid that moves inside a channel with two pipes. The color represent the norm of the velocity.

Theorems & Definitions (48)

  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3: Approximation with divergence-free virtual element functions
  • Lemma 3.4: Approximation property of $\Phi_h^k(\Omega_h)$
  • Remark 3.1
  • Proposition 4.1: Trace inequality
  • Proposition 4.2: Poicarè-Friedrichs, brenner-sung:2018
  • Proposition 4.3: Inverse estimate $H^2(E)-H^1(E)$
  • proof
  • Proposition 4.4: Inverse estimate $H^1(E)-L^2(E)$
  • ...and 38 more