Virtual element approximations of the time-fractional nonlinear convection-diffusion equation on polygonal meshes

Zaffar Mehdi Dar; M. Arrutselvi; Chandru Muthusamy; Sundararajan Natarajan; Gianmarco Manzini; Zaffar Mehdi Dar; M. Arrutselvi; Chandru Muthusamy; Sundararajan Natarajan; Gianmarco Manzini

doi:10.3934/mine.2025005

Mathematics in Engineering

2025, Volume 7, Issue 2: 96-129. doi: 10.3934/mine.2025005

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Research article

Virtual element approximations of the time-fractional nonlinear convection-diffusion equation on polygonal meshes

1.
Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore 632014, Tamil Nadu, India
2.
Department of Mechanical Engineering, Indian Institute of Technology, Madras, Chennai 600036, Tamil Nadu, India
3.
T-5 Group, Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico, USA

Received: 10 January 2025 Revised: 17 March 2025 Accepted: 20 March 2025 Published: 27 March 2025

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.
- Virtual Element Method,
- Grunwald-Letnikov approximation,
- regularity theory,
- projection operator,
- $ L^2- $norm,
- $ H^1- $norm,
- Sobolev space
Citation: Zaffar Mehdi Dar, M. Arrutselvi, Chandru Muthusamy, Sundararajan Natarajan, Gianmarco Manzini. Virtual element approximations of the time-fractional nonlinear convection-diffusion equation on polygonal meshes[J]. Mathematics in Engineering, 2025, 7(2): 96-129. doi: 10.3934/mine.2025005

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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.

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