Finite Element Analysis of Nonlinear 2D Space-Time-Fractional Diffusion Equations Using the $\hbox {L2-1}_{\sigma }$ Discretization Scheme

Abstract

This study proposes a high-precision finite element method (FEM) for nonlinear two-dimensional space-time-fractional diffusion equations, combining the Galerkin spatial discretization with the L2-$1_\sigma $ temporal scheme. By rigorously analyzing unconditional stability and deriving error estimates, the method achieves spatial convergence of order 2 and temporal convergence of order $3-\alpha $, where $\alpha \in (0,1)$, validated through numerical experiments under diverse initial conditions. Compared to existing works focusing on linear systems or single-term fractional dynamics, this research innovatively extends the L2-$1_\sigma $ scheme to complex nonlinear space-time-coupled problems, demonstrating its capability to simultaneously handle nonlinearities and fractional derivatives while maintaining computational efficiency and geometric adaptability.

Keywords

The multi-term fractional mixed diffusion and diffusion-wave equation / L2-$1_\sigma $ scheme / Finite element method (FEM) / H2N2 interpolation / Nonlinear / 65M10 / 78A48

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Funding

National Natural Science Foundation of China(12571388,12501543)

Natural Science Research Start-Up Foundation of Recruiting Talents of Nanjing University of Posts and Telecommunications(NY223127)

Science and Technology Innovative Research Team in Higher Educational Institutions of Hunan Province(2024RC3159)

the Project for the Natural Science Foundation of Hunan Province(2025JJ50035)

RIGHTS & PERMISSIONS

Shanghai University