Optimal order finite element approximations for variable-order time-fractional diffusion equations

Provides rigorous error analysis for numerical solutions of variable-order fractional PDEs, which are important in modeling anomalous diffusion but lack standard numerical guarantees.

The authors prove optimal convergence estimates for a fully discrete finite element method for variable-order time-fractional diffusion equations, achieving first-order accuracy in time and second order in space under mild regularity conditions.