A New $L2-1_σ$-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 $L2\!-\!1_\sigma$ 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 $\alpha_n\le \alpha(t_{n-\alpha_n/2})$, 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.