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Virtual element approximation of eigenvalue problems: is the stabilization of the right hand side necessary?

Daniele Boffi, Francesca Gardini, Lucia Gastaldi

Abstract

The VEM approximation of eigenvalue problems usually involves the appropriate tuning of stabilization parameters, unless self-stabilizing or stabilization-free VEM are used. In this paper we prove that for elliptic self-adjoint eigenvalue problems the stabilization of the mass matrix is not necessary when lower order standard VEM spaces are adopted. Numerical evidence shows that also for higher order schemes the same result is true on various mesh sequences.

Virtual element approximation of eigenvalue problems: is the stabilization of the right hand side necessary?

Abstract

The VEM approximation of eigenvalue problems usually involves the appropriate tuning of stabilization parameters, unless self-stabilizing or stabilization-free VEM are used. In this paper we prove that for elliptic self-adjoint eigenvalue problems the stabilization of the mass matrix is not necessary when lower order standard VEM spaces are adopted. Numerical evidence shows that also for higher order schemes the same result is true on various mesh sequences.

Paper Structure

This paper contains 12 sections, 10 theorems, 67 equations, 2 figures, 25 tables.

Table of Contents

  1. Introduction
  2. Abstract setting
  3. Property P1 for general VEM approximation of elliptic eigenvalue problem
  4. Virtual elements in practice
  5. Virtual element method
  6. Discrete eigenvalue problem and solution operator
  7. Convergence analysis
  8. Numerical results
  9. Triangular and square meshes
  10. Voronoi meshes
  11. Hexagonal meshes
  12. Dyadic meshes

Key Result

Lemma 1

The operator $\widetilde{T}_h$ maps the space $\mathbb{K}_{b,h}^\perp$ into itself. $\blacktriangleleft$$\blacktriangleleft$

Figures (2)

  • Figure 1: Coarsest meshes of different types: $\mathcal{T}$ triangles, $\mathcal{S}$ squares, $\mathcal{V}$ Voronoi, $\mathcal{H}$ hexagons
  • Figure 2: Coarsest dyadic mesh $\mathcal{D}$

Theorems & Definitions (17)

  • Lemma 1
  • proof
  • Theorem 2
  • Remark 1
  • Remark 2
  • Proposition 3
  • proof
  • Remark 3
  • Lemma 4
  • Lemma 5
  • ...and 7 more