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A stabilized virtual element method for singularly perturbed reaction–diffusion problem: A robustness analysis
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DOI：10.1016/j.cma.2026.119059.png)
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Abstract
En 中文
We propose and analyze a stabilized virtual element method (VEM) for the singularly perturbed reaction–diffusion equation, applicable to both reaction-dominated and diffusion-dominated regimes. The stabilization is introduced by incorporating a mesh-dependent term, derived from the least-squares formulation of the gradient of the residual associated with the diffusion–reaction equation. The theoretical analysis ensures the well-posedness of the stabilized discrete formulation, and provides a priori error estimates in the energy norm and L2-norm, demonstrating optimal convergence rates. Notably, the method shows full robustness in the energy norm with respect to the diffusion parameter, as the error estimate remains uniform even in the limit of vanishing diffusion. Additionally, the proposed method does not require tuning or scaling of the stabilization parameter. Several numerical examples are presented to illustrate the effectiveness and efficiency of the method and to validate the theoretically observed convergence rates.
Keywords:
virtual element method
singularly perturbed reaction–diffusion equation
stabilization
robustness
error estimates
Journal
IF:
7.3
Papers:
1.3W
Citations:
5.6W
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