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 $\mathbf{H}(\mathrm{div}, \Omega)$ 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.

翻译：本文针对二维和三维非线性应力辅助扩散问题的虚拟元离散化，建立并分析了基于残差的后验误差估计。该模型问题以扰动鞍点形式描述了弹性力学方程与扩散方程之间的双向耦合。通过建立鲁棒的全局 inf-sup 条件以及 $\mathbf{H}(\mathrm{div}, \Omega)$ 空间的 Helmholtz 分解，我们基于适当加权的范数构建了可靠且高效、能保证参数鲁棒性的误差估计器。后验误差分析采用了 Stokes 空间和边虚拟元空间的拟插值算子，为完备起见，文中给出了三维情形下该类算子的构造及其估计证明。最后，我们通过二维和三维数值实验验证了所提误差估计器的最优性能。