A New $L2-1_σ$-Interior Penalty Method for Variable-Order Time-Fractional Subdiffusion Interface Problem with Curved Interface

Abstract: This paper treats variable-order time-fractional subdiffusion with discontinuous coefficients across a curved interface using $L2!-!1_σ$ 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 $αn$ at $t{n-σn}$ lies in the range of $α(t)$ on $[t{n-1},t_n]$ and $αn\le α(t{n-α_n/2})$, experiments indicate the second inequality can be relaxed or omitted, enabling straightforward selection of $α_n$ from many admissible values without solving a nonlinear equation. Numerical results verify temporal rates $\min{2,rδ}$, spatial order $\min{s,k+1}$, and robustness to superconvergent points and interface geometry.