用于耦合应力辅助扩散问题的鲁棒虚拟元方法
Robust virtual element methods for coupled stress-assisted diffusion problems
AI总结：
该研究针对应力辅助扩散耦合问题，完成了混合非线性公式的鲁棒连续分析，提出虚拟元离散方案并证明其适定性与先验误差估计，数值实验验证了方法的精度与性能。
摘要：
本文首先旨在对与弹性材料相互作用的溶质的应力辅助扩散的混合非线性公式进行鲁棒的连续分析，其次提出并分析该模型问题的虚拟元公式。线性弹性的Herrmann公式与反应-扩散方程（以混合形式书写）之间的双向耦合机制由扩散诱导的主动应力和应力相关的扩散构成。我们采用针对扰动鞍点问题的扩展Babuška–Brezzi–Braess理论对这两个子问题进行分析。在数据较小的假设下，通过Banach不动点策略证明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的虚拟元公式具有唯一解。我们推导了先验误差估计，并通过计算仿真测试了所提方法的精度和性能。
英文摘要：
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 Babuška--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.
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