Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the stabilization of a virtual element method for an acoustic vibration problem

Linda Alzaben, Daniele Boffi, Andreas Dedner, Lucia Gastaldi

TL;DR

This work develops a rigorous framework for the convergence of a virtual element method for an acoustic vibration eigenproblem, examining how stabilization influences spectral accuracy. By recasting the problem in a mixed Kikuchi-type formulation and leveraging the discrete compactness property, the authors establish conditions under which stabilized VEM schemes converge and, in some cases, remain stabilization-free, notably for the lowest-order triangular elements. They prove EDk/WA/SA criteria imply uniform convergence of the discrete solution operator to the continuous one and demonstrate that stabilization can be unnecessary in several practical scenarios. Numerical experiments on rectangular and L-shaped domains corroborate the theory, showing parameter-free accuracy for certain meshes and orders, while highlighting mesh-dependent behavior and instances where stabilization restores optimal convergence.

Abstract

In this paper we introduce an abstract setting for the convergence analysis of the virtual element approximation of an acoustic vibration problem. We discuss the effect of the stabilization parameters and remark that in some cases it is possible to achieve optimal convergence without the need of any stabilization. This statement is rigorously proved for lowest order triangular element and supported by several numerical experiments.

On the stabilization of a virtual element method for an acoustic vibration problem

TL;DR

This work develops a rigorous framework for the convergence of a virtual element method for an acoustic vibration eigenproblem, examining how stabilization influences spectral accuracy. By recasting the problem in a mixed Kikuchi-type formulation and leveraging the discrete compactness property, the authors establish conditions under which stabilized VEM schemes converge and, in some cases, remain stabilization-free, notably for the lowest-order triangular elements. They prove EDk/WA/SA criteria imply uniform convergence of the discrete solution operator to the continuous one and demonstrate that stabilization can be unnecessary in several practical scenarios. Numerical experiments on rectangular and L-shaped domains corroborate the theory, showing parameter-free accuracy for certain meshes and orders, while highlighting mesh-dependent behavior and instances where stabilization restores optimal convergence.

Abstract

In this paper we introduce an abstract setting for the convergence analysis of the virtual element approximation of an acoustic vibration problem. We discuss the effect of the stabilization parameters and remark that in some cases it is possible to achieve optimal convergence without the need of any stabilization. This statement is rigorously proved for lowest order triangular element and supported by several numerical experiments.
Paper Structure (25 sections, 17 theorems, 138 equations, 7 figures, 14 tables)

This paper contains 25 sections, 17 theorems, 138 equations, 7 figures, 14 tables.

Table of Contents

  1. Introduction
  2. Function spaces and preliminaries
  3. The continuous problem
  4. Problem setting and its variational formulation
  5. Mixed formulation
  6. The virtual element discretization
  7. The discrete variational formulation
  8. Discrete mixed formulation
  9. Spectral approximation and convergence analysis
  10. Approximation properties of the VEM space and other preliminary results
  11. Consistency
  12. Uniform convergence of $T_h$ to $T$
  13. Discrete Compactness Property
  14. SDCP implies EDK, WA and SA
  15. Stabilized VEM spaces
  16. ...and 10 more sections

Key Result

Lemma 1

If $\Omega$ is a polygonal domain, then there exists $s\in(1/2,1)$, such that the subspace $\mathbf{H}_0(\mathop{\mathrm{\mathrm{div}}}\nolimits;\Omega)\cap\mathbf{H}(\mathop{\mathrm{\mathrm{rot}}}\nolimits^0;\Omega)$ is contained in $\mathbf{H}^s(\Omega)$ which is compactly embedded into $\mathbf{L

Figures (7)

  • Figure 1: Domain with subregions
  • Figure 2: Zoom of the interior region $\Omega_I$ with the vectors
  • Figure 3: Zoom of the right upper corner of $\Omega$ with the vectors
  • Figure 4: Examples of triangular, square, trapezoidal meshes: $\mathcal{T}_h$ left, $\mathcal{Q}_h$ middle and $\mathcal{Z}_h$ right, corresponding to level $\ell=0$
  • Figure 5: Examples of meshes: $\mathcal{V}_h$(left) and $\mathcal{H}_h$(right) corresponding to level $\ell=0$
  • ...and 2 more figures

Theorems & Definitions (43)

  • Lemma 1
  • Remark 1
  • Lemma 2
  • Proposition 1
  • proof
  • Remark 2
  • Remark 3
  • Proposition 2
  • proof
  • Lemma 3
  • ...and 33 more