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A noncoforming virtual element approximation for the Oseen eigenvalue problem

Dibyendu Adak, Felipe Lepe, Gonzalo Rivera

TL;DR

This work develops a divergence‑free, nonconforming virtual element method for the two‑dimensional Oseen eigenvalue problem. By formulating the problem in an operator framework using a compact solution operator on $L^2$, the authors prove convergence and establish spectral error estimates that yield double order convergence for eigenvalues. The NCVEM uses polygonal meshes with a divergence‑preserving discretization and carefully designed local/projection operators to maintain incompressibility and stability. They demonstrate, both theoretically and numerically, that spurious eigenvalues can be controlled and that optimal convergence rates are recovered on convex domains, with expected reductions on domains with singularities. The results have practical implications for efficiently solving nonsymmetric fluid‑mechanics eigenproblems on general polygonal meshes, with stabilization choices influencing spurious modes and accuracy.

Abstract

In this paper we analyze a nonconforming virtual element method to approximate the eigenfunctions and eigenvalues of the two dimensional Oseen eigenvalue problem. The spaces under consideration lead to a divergence-free method which is capable to capture properly the divergence at discrete level and the eigenvalues and eigenfunctions. Under the compact theory for operators we prove convergence and error estimates for the method. By employing the theory of compact operators we recover the double order of convergence of the spectrum. Finally, we present numerical tests to assess the performance of the proposed numerical scheme.

A noncoforming virtual element approximation for the Oseen eigenvalue problem

TL;DR

This work develops a divergence‑free, nonconforming virtual element method for the two‑dimensional Oseen eigenvalue problem. By formulating the problem in an operator framework using a compact solution operator on , the authors prove convergence and establish spectral error estimates that yield double order convergence for eigenvalues. The NCVEM uses polygonal meshes with a divergence‑preserving discretization and carefully designed local/projection operators to maintain incompressibility and stability. They demonstrate, both theoretically and numerically, that spurious eigenvalues can be controlled and that optimal convergence rates are recovered on convex domains, with expected reductions on domains with singularities. The results have practical implications for efficiently solving nonsymmetric fluid‑mechanics eigenproblems on general polygonal meshes, with stabilization choices influencing spurious modes and accuracy.

Abstract

In this paper we analyze a nonconforming virtual element method to approximate the eigenfunctions and eigenvalues of the two dimensional Oseen eigenvalue problem. The spaces under consideration lead to a divergence-free method which is capable to capture properly the divergence at discrete level and the eigenvalues and eigenfunctions. Under the compact theory for operators we prove convergence and error estimates for the method. By employing the theory of compact operators we recover the double order of convergence of the spectrum. Finally, we present numerical tests to assess the performance of the proposed numerical scheme.

Paper Structure

This paper contains 16 sections, 11 theorems, 116 equations, 9 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. The variational formulation
  4. The virtual element method
  5. Mesh notation and mesh regularity
  6. Local and global discrete space
  7. The discrete eigenvalue problem
  8. A priori error estimates for the source problem
  9. $\bold{L}^2$ Error estimates for the velocity
  10. Spectral approximation and error estimates:
  11. Numerical experiments
  12. Test 1: a square domain
  13. Test case 2: $\mathrm{L}$ shaped domain
  14. Spurious analysis
  15. Effect of the stabilization
  16. ...and 1 more sections

Key Result

Theorem 2.1

\newlabelth:regularidadfuente0 There exists $s>0$ such that for all $\boldsymbol{f} \in \bold{L}^2(\Omega,\mathbb{C})$, 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 \bold{H}^{1+ where $C := \dfrac{C_{pf}}{\beta}\max\left\lbrace 1, \dfrac{C_{pf}\|\boldsymbol{\beta}\|_{\infty,\Om

Figures (9)

  • Figure 1: \newlabelfig:mesh0 Sample meshes: ${\mathcal{T}}_h^1$ (top left), ${\mathcal{T}}_h^2$ (top right), ${\mathcal{T}}_h^3$ (bottom left), ${\mathcal{T}}_h^4$ (bottom right) for $N=8$ and 10.
  • Figure 2: First, second and third magnitude of the eigenfunctions in the square together with the associated pressures: first column $u_{1,h}$, $u_{2,h}$ and $u_{3,h}$ ;second column: $p_{1,h}$, $p_{2,h}$ and $p_{3,h}$; for different family of meshes.
  • Figure 3: \newlabelfig:mesh:L0 Sample meshes: ${\mathcal{T}}_h^5$ ( left panel), ${\mathcal{T}}_h^6$ ( right panel) for $N=8$
  • Figure 4: First, second, and third magnitude of the eigenfunctions in the nonconvex $L$ domain together with the associated pressures: first column $u_{1,h}$, $u_{2,h}$ and $u_{3,h}$ ;second column: $p_{1,h}$, $p_{2,h}$ and $p_{3,h}$; for different family of meshes.
  • Figure 5: First, second, and third magnitude of the eigenfunctions with $N=32$, for different family of meshes.
  • ...and 4 more figures

Theorems & Definitions (15)

  • Theorem 2.1
  • Lemma 2.2
  • Theorem 2.3
  • Lemma 2.4
  • Remark 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Remark 3.5
  • Theorem 4.1
  • Proof 1
  • ...and 5 more