A New $L2-1_{\sigma }$-Interior Penalty Method for Variable-Order Time-Fractional Subdiffusion Interface Problem with Curved Interface

This paper treats variable-order time-fractional subdiffusion with discontinuous coefficients across a curved interface using $L2\text{\hspace{0.17em}⁣}-\text{\hspace{0.17em}⁣}{1}_{\sigma }$ time stepping on graded meshes and a symmetric interior penalty FEM on body-fitted meshes. Stability and optimal a priori error estimates in a discrete-in-time $L^2$ norm are established, yielding second-order temporal accuracy. While analysis typically assumes ${\alpha }_n$ at $t_{n-{\sigma }_n}$ lies in the range of $\alpha (t)$ on $[t_{n-1},t_n]$ and ${\alpha }_n\le \alpha (t_{n-{\alpha }_n\mathrm{/}2})$, experiments indicate the second inequality can be relaxed or omitted, enabling straightforward selection of ${\alpha }_n$ from many admissible values without solving a nonlinear equation. Numerical results verify temporal rates $\min \{2,r\delta \}$, spatial order $\min \{s,k+1\}$, and robustness to superconvergent points and interface geometry.