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Interior penalty discontinuous Galerkin methods for the nearly incompressible elasticity eigenvalue problem with heterogeneous media

Arbaz Khan, Felipe Lepe, Jesus Vellojin

TL;DR

This work develops interior penalty discontinuous Galerkin methods for the Herrmann formulation of the nearly incompressible elasticity eigenvalue problem in heterogeneous media. By employing a weighted Lam\'e norm and noncompact-operator theory, it proves convergence of both the continuous and discrete spectra as the mesh is refined, uniformly in the Lam\'e parameters, and establishes a residual-based a posteriori estimator that is robust as $\lambda\to\infty$ (Stokes limit). The authors provide a complete a priori and a posteriori analysis, including stability, error estimates, and spectral convergence, and validate the approach with extensive 2D and 3D numerical experiments showing optimal rates for symmetric IPDG and reliable adaptivity near singularities and material interfaces. The results advance robust, high-fidelity spectral approximations for complex elasticity problems in heterogeneous media and offer practical guidance on stabilization parameter selection and adaptive refinement strategies.

Abstract

This paper studies the family of interior penalty discontinuous Galerkin methods for solving the Herrmann formulation of the linear elasticity eigenvalue problem in heterogeneous media. By employing a weighted Lamé coefficient norm within the framework of non-compact operators theory, we prove convergence of both continuous and discrete eigenvalue problems as the mesh size approaches zero, independently of the Lamé constants. Additionally, we conduct an a posteriori analysis and propose a reliable and efficient estimator. The theoretical findings are supported by numerical experiments.

Interior penalty discontinuous Galerkin methods for the nearly incompressible elasticity eigenvalue problem with heterogeneous media

TL;DR

This work develops interior penalty discontinuous Galerkin methods for the Herrmann formulation of the nearly incompressible elasticity eigenvalue problem in heterogeneous media. By employing a weighted Lam\'e norm and noncompact-operator theory, it proves convergence of both the continuous and discrete spectra as the mesh is refined, uniformly in the Lam\'e parameters, and establishes a residual-based a posteriori estimator that is robust as (Stokes limit). The authors provide a complete a priori and a posteriori analysis, including stability, error estimates, and spectral convergence, and validate the approach with extensive 2D and 3D numerical experiments showing optimal rates for symmetric IPDG and reliable adaptivity near singularities and material interfaces. The results advance robust, high-fidelity spectral approximations for complex elasticity problems in heterogeneous media and offer practical guidance on stabilization parameter selection and adaptive refinement strategies.

Abstract

This paper studies the family of interior penalty discontinuous Galerkin methods for solving the Herrmann formulation of the linear elasticity eigenvalue problem in heterogeneous media. By employing a weighted Lamé coefficient norm within the framework of non-compact operators theory, we prove convergence of both continuous and discrete eigenvalue problems as the mesh size approaches zero, independently of the Lamé constants. Additionally, we conduct an a posteriori analysis and propose a reliable and efficient estimator. The theoretical findings are supported by numerical experiments.
Paper Structure (19 sections, 19 theorems, 98 equations, 7 figures, 6 tables)

This paper contains 19 sections, 19 theorems, 98 equations, 7 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Notations
  3. Model problem
  4. Main contribution
  5. Outline
  6. Variational formulation and preliminary results
  7. The DG method
  8. Preliminaries
  9. Symmetric and nonsymmetric DG schemes
  10. Convergence and error estimates
  11. A posteriori error analysis
  12. The local and global indicators
  13. Reliability
  14. Efficiency
  15. Numerical experiments
  16. ...and 4 more sections

Key Result

Lemma 2.1

Let $(\widehat{\boldsymbol{u}},\widehat{p})$ be the unique solution of def:system_source_complete, then there exists $s\in(0,1]$ such that $(\widehat{\boldsymbol{u}},\widehat{p})\in \mathrm{H}^{1+s}(\Omega)^{d}\times \mathrm{H}^{s}(\Omega)$ and the following estimate holds (see for instance grisvard where the hidden constant depends on $\Omega$ and the Lamé coefficients.

Figures (7)

  • Figure 1: Test \ref{['subsec:square2D_two_domain']}. Left: Sample mesh for the subdivided computational domain with two materials. The next two figures correspond to the warped domain using the lowest mode $\boldsymbol{u}_{1,h}$ for $\nu=0.35$ and $\nu=0.5$, respectively. The right figure correspond to the first pressure mode, which is similar for $\nu=0.35$ and $\nu=0.5$.
  • Figure 2: Test \ref{['subsec:square2D_two_domain']}. Intermediate meshes of the square domain with different materials obtained with the adaptive algorithm and different values of $\nu$, with $k=1,2$.
  • Figure 3: Test \ref{['subsec:square2D_two_domain']}. Error curves obtained from the adaptive algorithm in the square domain compared with the lines $\mathcal{O}(\texttt{dof}^{-1})$ and $\mathcal{O}(\texttt{dof}^{-2})$.
  • Figure 4: Test \ref{['subsec:square2D_two_domain']}. Estimator and efficiency curves obtained from the adaptive algorithm in the square domain with different values of $\nu$ and $k=1,2$.
  • Figure 5: Test \ref{['subsec:Lshape3D-domain']}. Initial mesh for the subdivided computational domain $\Omega:=\Omega_1\cup\Omega_2$ (left) together with a comparison of the first displacement eigenmodes for $\nu=0.35$ (middle) and $\nu=0.5$ (right) at the last adaptive iteration.
  • ...and 2 more figures

Theorems & Definitions (21)

  • Lemma 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Lemma 2.4
  • Remark 3.1
  • Lemma 3.2: ellipticity of $a_h(\cdot,\cdot)$
  • Lemma 4.1
  • Corollary 4.2
  • Lemma 4.3
  • Lemma 4.4
  • ...and 11 more