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ROBUST VIRTUAL ELEMENT METHODS FOR COUPLED STRESS-ASSISTED DIFFUSION PROBLEMS
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DOI：10.1137/24M163640X.png)
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Abstract
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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 analyze 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 subproblems are analyzed using the extended Babusv \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 subproblems 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.
Keywords:
virtual element methods
stress-assisted diffusion
diffusion-induced stress
perturbed saddle-point problems
fixed-point operators
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