A New $L2-1_σ$-Interior Penalty Method for Variable-Order Time-Fractional Subdiffusion Interface Problem with Curved Interface
It provides a stable and optimally convergent numerical scheme for a challenging class of fractional PDEs with discontinuous coefficients and curved interfaces, though the contribution is incremental as it extends existing methods to variable-order and curved interfaces.
The paper develops an L2-1σ interior penalty method for variable-order time-fractional subdiffusion with a curved interface, achieving second-order temporal accuracy and optimal spatial convergence, with numerical verification of rates min{2, rδ} in time and min{s, k+1} in space.
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 $α_n$ at $t_{n-σ_n}$ lies in the range of $α(t)$ on $[t_{n-1},t_n]$ and $α_n\le α(t_{n-α_n/2})$, 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.