A posteriori analysis of a virtual element approach on polytopal meshes for the buckling eigenvalue problem

Abstract

We introduce a novel residual-based a posteriori error estimator for the conforming $C^1$ Virtual Element Method (VEM) applied to the buckling eigenvalue problem, incorporating nonlinear plane stress effects in both two and three dimensions. The estimator is fully computable on general polyhedral meshes and implemented within the open-source \texttt{vem++} library. Its reliability is rigorously justified via bounds on the residual equation using polynomial projections, stabilisation contributions, and interpolation estimates, while efficiency is ensured through the use of bubble function arguments. Comprehensive numerical experiments in 2D and 3D illustrate the estimator's optimal accuracy and robustness, highlighting its potential for predictive analysis of complex plate structures.

Figures (8)

• Figure 1: Example 1. Polytopal discretisations used for the uniform refinement test.
• Figure 2: Example 2. Polytopal discretisations used for the adaptive refinement test.
• Figure 3: Example 2. Snapshots of the polynomial projection of the first three eigenfunctions in the last refinement step for the L-shaped mesh with $\kappa_0=0$.
• Figure 4: Example 2. Snapshots of the polynomial projection of the first three eigenfunctions in the last refinement step for the perfored circle mesh with $\kappa_0=\frac{2}{3}$.
• Figure 5: Example 2. Snapshots of the polynomial projection of the first three eigenfunctions in the last refinement step for the perfored Fichera cube mesh with $\kappa_0=1$, the isosurfaces are computed from paraview starting from the mean value of the projected virtual function on each vertex.

Theorems & Definitions (31)

• Theorem 2.1
• Proposition 3.1: polynomial approximation
• Remark 3.1
• Lemma 3.1
• Lemma 3.2
• proof
• Theorem 3.1
• proof
• Lemma 3.3
• proof