This paper aims first to perform robust continuous analysis of a mixed nonlinear formulation for stress-assisted diffusion of a solute that interacts with an elastic material, and second to propose and analyse a virtual element formulation of the model problem. The two-way coupling mechanisms between the Herrmann formulation for linear elasticity and the reaction-diffusion equation (written in mixed form) consist of diffusion-induced active stress and stress-dependent diffusion. The two sub-problems are analysed using the extended Babu\v{s}ka--Brezzi--Braess theory for perturbed saddle-point problems. The well-posedness of the nonlinearly coupled system is established using a Banach fixed-point strategy under the smallness assumption on data. The virtual element formulations for the uncoupled sub-problems are proven uniquely solvable by a fixed-point argument in conjunction with appropriate projection operators. We derive the a priori error estimates, and test the accuracy and performance of the proposed method through computational simulations.

翻译：本文首先对溶质与弹性材料相互作用下的应力辅助扩散问题的混合非线性形式进行鲁棒连续分析，其次提出并分析该模型问题的虚拟元离散格式。线性弹性的Herrmann格式与反应-扩散方程（以混合形式表述）之间的双向耦合机制包括扩散诱导的主动应力和应力依赖的扩散。两个子问题采用针对扰动鞍点问题的扩展Babuška-Brezzi-Braess理论进行分析。在数据充分小的假设下，通过Banach不动点策略建立了非线性耦合系统的适定性。通过不动点论证结合适当的投影算子，证明了非耦合子问题虚拟元格式的唯一可解性。我们推导了先验误差估计，并通过数值模拟测试了所提方法的精度与性能。