A posteriori error estimation for an interior penalty virtual element method of Kirchhoff plates

TL;DR

This work develops a residual-type a posteriori error estimator for an interior penalty virtual element method (IPVEM) applied to the Kirchhoff plate bending problem on polygonal meshes. Central to the approach is a modified discrete formulation that incorporates the $H^1$-elliptic projector in the jump and average terms, enabling computable estimators and simplifying implementation. The authors construct an enriching operator to prove reliability and efficiency of the estimator and show that IPVEM behaves like a $C^0$-continuous method (quasi-$C^0$ continuity), which allows omitting the jumps of the discrete function in the estimator. An adaptive VEM with one-hanging-node refinement is proposed, and numerical experiments across smooth, large-gradient, and low-regularity scenarios confirm robustness and optimal convergence of the estimator and the adaptive scheme.

Abstract

In this paper, we develop a residual-type a posteriori error estimation for an interior penalty virtual element method (IPVEM) for the Kirchhoff plate bending problem. Building on the work in \cite{FY2023IPVEM}, we adopt a modified discrete variational formulation that incorporates the $ H^1 $-elliptic projector in the jump and average terms. This allows us to simplify the numerical implementation by including the $ H^1 $-elliptic projector in the computable error estimators. We derive the reliability and efficiency of the a posteriori error bound by constructing an enriching operator and establishing some related error estimates that align with $C^0$-continuous interior penalty finite element methods. As observed in the a priori analysis, the interior penalty virtual elements exhibit similar behaviors to $C^0$-continuous elements despite its discontinuity. This observation extends to the a posteriori estimate since we do not need to account for the jumps of the function itself in the discrete scheme and the error estimators. As an outcome of the error estimator, an adaptive VEM is introduced by means of the mesh refinement strategy with the one-hanging-node rule. Numerical results from several benchmark tests confirm the robustness of the proposed error estimators and show the efficiency of the resulting adaptive VEM.

Figures (8)

• Figure 1: The final adapted mesh and the exact and numerical solutions with homogeneous boundary values conditions
• Figure 2: (a) The estimator $\eta(u_h, \mathcal{T}_h)$ for adapted meshes; (b) The convergence orders
• Figure 3: The final adapted mesh and the exact and numerical solutions with inhomogeneous boundary values conditions
• Figure 4: The initial and the final adapted meshes. (a) The initial mesh; (b) After 10 refinement steps; (c) After 20 refinement steps; (d) The zoomed mesh in (c)
• Figure 5: Convergence rates of the error $|u - \Pi_h^\Delta u_h|_{2,h}$ and the error estimator $\eta(u_h)$

Theorems & Definitions (23)

• Lemma 2.1
• Lemma 3.1: Inverse inequality
• Lemma 3.2
• Lemma 3.3
• proof
• Remark 3.1
• Lemma 3.4
• proof
• Lemma 4.1
• proof