@article{Dar2025, 
author = {Zaffar Mehdi Dar and M. Arrutselvi and Chandru Muthusamy and Sundararajan Natarajan and Gianmarco Manzini},
title = {Virtual element approximations of the time-fractional nonlinear convection-diffusion equation on polygonal meshes},
year = {2025},
journal = {Mathematics in Engineering},
volume = {7},
number = {2},
pages = {96-129},
keywords = {Virtual Element Method, Grunwald-Letnikov approximation, regularity theory, projection operator, L2−norm, H1−norm, Sobolev space},
url = {https://www.sciopen.com/article/10.3934/mine.2025005},
doi = {10.3934/mine.2025005},
abstract = {We extend the Virtual Element Method to a two-dimensional unsteady nonlinear convection-diffusion equation characterized by a fractional-order derivative with respect to the time variable. Our methodology is based on three fundamental technical components: a fractional version of the Grunwald-Letnikov approximation, discrete maximal regularity, and the regularity theory associated with non-linearity. We prove the method's well-posedness, i.e., the approximate solution's existence and uniqueness to the time-fractional convection-diffusion equation with a Lipschitz nonlinear source term. The fully discrete scheme inherently maintains stability and consistency by leveraging the discrete maximal regularity and the energy projection operator. The convergence in the        L    2  -norm and        H    1  -norm to various mesh configurations is validated by numerical results, underlining the practical effectiveness of the proposed method.}
}