A New $L2-1_{\ddot{I}}ƒ$-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 $L2-1_{\ddot{I}}ƒ$ 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 $\hat{I}{\pm }_n$ at $t_{n-\ddot{I}{ƒ}_n}$ lies in the range of $\hat{I}\pm (t)$ on $[t_{n-1},t_n]$ and $\hat{I}{\pm }_n\le \hat{I}\pm (t_{n-\hat{I}{\pm }_n/2})$, experiments indicate the second inequality can be relaxed or omitted, enabling straightforward selection of $\hat{I}{\pm }_n$ from many admissible values without solving a nonlinear equation. Numerical results verify temporal rates $\min \{2,r\hat{I}´\}$, spatial order $\min \{s,k+1\}$, and robustness to superconvergent points and interface geometry.

18 pages, 1 figure