A posteriori error analysis of a robust virtual element method for stress-assisted diffusion problems

We develop and analyse residual-based a posteriori error estimates for the virtual element discretisation of a nonlinear stress-assisted diffusion problem in two and three dimensions. The model problem involves a two-way coupling between elasticity and diffusion equations in perturbed saddle-point form. A robust global inf-sup condition and Helmholtz decomposition for $\mathbf{H}(\mathrm{div}, Ω)$ lead to a reliable and efficient error estimator based on appropriately weighted norms that ensure parameter robustness. The a posteriori error analysis uses quasi-interpolation operators for Stokes and edge virtual element spaces, and we include the proofs of such operators with estimates in 3D for completeness. Finally, we present numerical experiments in both 2D and 3D to demonstrate the optimal performance of the proposed error estimator.

💡 Research Summary

This paper presents a comprehensive residual‑based a posteriori error analysis for the virtual element method (VEM) applied to a nonlinear stress‑assisted diffusion problem in both two and three spatial dimensions. The governing equations consist of a two‑way coupling between linear elasticity and a diffusion equation, forming a perturbed saddle‑point system. The authors first reformulate the problem in a mixed‑mixed setting and introduce parameter‑dependent weighted norms that incorporate the Lamé parameters, the diffusion coefficient, and the model parameter θ. These norms are crucial for achieving robustness with respect to large variations in physical parameters.

A global inf‑sup condition is proved for the coupled system, showing that the stability constants are independent of the material parameters. This result, together with the Babuška‑Brezzi theory, guarantees well‑posedness under a small‑data assumption arising from the fixed‑point treatment of the nonlinear terms.

The discretisation follows the VEM framework introduced in earlier work