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A Conforming virtual element approximation for the Oseen eigenvalue problem

Danilo Amigo, Felipe Lepe, Nitesh Verma

TL;DR

This work develops a conforming virtual element method for the 2D Oseen eigenvalue problem in a velocity–pressure formulation, using divergence-conforming VEM spaces on polygonal meshes. By constructing stable and consistent discrete sesquilinear forms and proving the compactness of the continuous and discrete solution operators ($\boldsymbol{T}$ and $\boldsymbol{T}_h$), the authors establish spurious-free spectral convergence with eigenvalue errors of order $h^{\sigma+\sigma^*}$ and $L^2$-velocity errors via a duality argument. The analysis leverages compact-operator theory and dual problems to address the non-selfadjoint nature of the Oseen operator, ensuring robust convergence results. Numerical experiments on convex and non-convex domains validate the theoretical rates and illustrate stabilization effects on spurious eigenvalues, highlighting practical guidance for mesh design and stabilization in polygonal-VEM eigenvalue computations.

Abstract

In this paper we analyze a conforming virtual element method to approximate the eigenfunctions and eigenvalues of the two dimensional Oseen eigenvalue problem. We consider the classic velocity-pressure formulation which allows us to consider the divergence-conforming virtual element spaces employed for the Stokes equations. Under standard assumptions on the meshes we derive a priori error estimates for the proposed method with the aid of the compact operators theory. We report some numerical tests to confirm the theoretical results.

A Conforming virtual element approximation for the Oseen eigenvalue problem

TL;DR

This work develops a conforming virtual element method for the 2D Oseen eigenvalue problem in a velocity–pressure formulation, using divergence-conforming VEM spaces on polygonal meshes. By constructing stable and consistent discrete sesquilinear forms and proving the compactness of the continuous and discrete solution operators ( and ), the authors establish spurious-free spectral convergence with eigenvalue errors of order and -velocity errors via a duality argument. The analysis leverages compact-operator theory and dual problems to address the non-selfadjoint nature of the Oseen operator, ensuring robust convergence results. Numerical experiments on convex and non-convex domains validate the theoretical rates and illustrate stabilization effects on spurious eigenvalues, highlighting practical guidance for mesh design and stabilization in polygonal-VEM eigenvalue computations.

Abstract

In this paper we analyze a conforming virtual element method to approximate the eigenfunctions and eigenvalues of the two dimensional Oseen eigenvalue problem. We consider the classic velocity-pressure formulation which allows us to consider the divergence-conforming virtual element spaces employed for the Stokes equations. Under standard assumptions on the meshes we derive a priori error estimates for the proposed method with the aid of the compact operators theory. We report some numerical tests to confirm the theoretical results.
Paper Structure (12 sections, 19 theorems, 123 equations, 5 figures, 7 tables)

This paper contains 12 sections, 19 theorems, 123 equations, 5 figures, 7 tables.

Table of Contents

  1. Introduction
  2. The variational formulation
  3. The virtual element method
  4. The discrete sesquilinear forms
  5. Technical approximation results
  6. The dual discrete eigenvalue problem
  7. Error estimates
  8. Error in $\mathrm{L}^2$ norm
  9. Numerical experiments
  10. Test 1: Convex domain
  11. L-shaped Domain
  12. Effects of the stabilization in the computed spectrum

Key Result

Theorem 2.1

\newlabelth:regularidadfuente0 There exists $s>0$ such that for all $\boldsymbol{f} \in \mathrm{H}_0^1(\Omega,\mathbb{C})^2$, the solution $(\widehat{\boldsymbol{u}},\widehat{p})\in\mathcal{X}$ of problem def:oseen_system_weak_source, satisfies for the velocity $\widehat{\boldsymbol{u}}\in \mathr where $C := \dfrac{C_{pf}}{\beta}\max\left\lbrace 1, \dfrac{C_{pf}\|\boldsymbol{\beta}\|_{\infty,\Om

Figures (5)

  • Figure 1: \newlabelfig:mesh0 Sample meshes: ${\mathcal{T}}_h^1$ (top left), ${\mathcal{T}}_h^2$ (top center), ${\mathcal{T}}_h^3$ (top right), ${\mathcal{T}}_h^4$ (bottom left), ${\mathcal{T}}_h^5$ (bottom right).
  • Figure 2: \newlabelfig:eigs10 First, second and third magnitude of the eigenfunctions and their associated pressures. First column: $\boldsymbol{u}_{h1}$, $\boldsymbol{u}_{h2}$ and $\boldsymbol{u}_{h3}$. Second column: $p_{h1}$, $p_{h2}$ and $p_{h3}$.
  • Figure 3: \newlabelfig:Ldomain0 Sample of mesh ${\mathcal{T}}_h^{6}$.
  • Figure 4: \newlabelfig:eigs20 First, second and third magnitude of the eigenfunctions and their associated pressures for the L-shaped domain. First column: $\boldsymbol{u}_{h1}$, $\boldsymbol{u}_{h2}$ and $\boldsymbol{u}_{h3}$. Second column: $p_{h1}$, $p_{h2}$ and $p_{h3}$.
  • Figure 5: \newlabelfigesp0 Magnitude of the first four eigenfunctions. From left to right: $\boldsymbol{u}_{h1}$ and $\boldsymbol{u}_{h2}$ (top); $\boldsymbol{u}_{h3}$ and $\boldsymbol{u}_{h4}$ (bottom).

Theorems & Definitions (27)

  • Theorem 2.1
  • Lemma 2.2
  • Theorem 2.3
  • Lemma 2.4
  • Remark 2.5
  • Remark 3.1
  • Lemma 3.2: Existence of a virtual approximation operator
  • Lemma 3.3: Existence of an interpolation operator
  • Lemma 3.4
  • Lemma 3.5
  • ...and 17 more