Optimal Error Estimates of a Fast C-Bézier Finite Element Method for Time-Fractional Anomalous Transport in Heterogeneous Media

Time-fractional diffusion equations (TFDEs) are essential for modeling anomalous transport in heterogeneous media, but high-fidelity long-time simulations face two bottlenecks: the O(N2) complexity of non-local fractional derivatives, and the spatial truncation error of polynomial-based finite element methods (FEMs) when resolving oscillatory plumes or singular sources. We propose a framework combining a C-Bézier FEM for spatial approximation with a fast L1 temporal discretization. By coupling the shape parameter of the C-Bézier basis to the mesh size (μ=πh), the scheme reproduces trigonometric profiles of the corresponding frequency exactly; for solutions whose spatial part lies in the C-Bézier space this eliminates the spatial truncation error and drives the associated error constant to near zero. A sum-of-exponentials (SOE) approximation reduces the temporal complexity from O(N2) to O(N) and storage to O(1), enabling scalable 3D simulation. We prove the optimal O(τ2−α+hk+1) convergence, and numerical experiments confirm these rates. For profiles matched by the basis, the method yields substantially smaller errors than Lagrange FEM; for a general solution outside the C-Bézier space, the two methods share the same order and comparable error magnitudes, so the gains are specific to fields reproduced by the basis. We further examine low-regularity scenarios, including discontinuous interfaces and Dirac-delta injections.