Robust virtual element methods for 3D stress-assisted diffusion problems

TL;DR

This paper develops a robust virtual element method for accurately solving complex 3D stress-assisted diffusion problems, ensuring stability and convergence through advanced mathematical analysis.

Contribution

It introduces a novel VEM framework with specialized operators for coupled non-linear 3D diffusion and stress problems, backed by rigorous theoretical analysis.

Findings

Demonstrates high accuracy in numerical simulations

Shows robustness and stability of the proposed method

Applicable to complex 3D stress-diffusion scenarios

Abstract

This paper presents an initial exploration of stress-assisted diffusion problems in three dimensions within the framework of the virtual element method (VEM). Hilbert spaces enriched with parameter-weighted norms, the extended Babu\v{s}ka-Brezzi-Braess theory for perturbed saddle-point problems, and Banach fixed-point theory play a crucial role in performing a robust analysis of the fully coupled non-linear system. The proposed virtual element formulations are provided with appropriate projection, interpolation, and stabilisation operators that ensures the well-posedness of the discrete problem. Numerical simulations are conducted to show the accuracy, performance, and applicability of the method.