A Predictor‐Corrector Linearized High‐Order FEM for Nonlinear Time‐Fractional Parabolic Equations With Distributed Delay and Variable Coefficients

ABSTRACT This paper presents a comprehensive numerical framework for solving nonlinear time‐fractional parabolic equations with distributed delays and variable coefficients. The proposed method combines the high‐order Alikhanov ‐ temporal discretization on graded meshes with a novel predictor‐corrector quasi‐linearization technique for handling the coupled nonlinearity and delay terms. Spatial discretization is achieved through the standard Galerkin finite element method. We establish the unconditional stability of the scheme and derive optimal error estimates of order in the ‐norm and in the ‐norm. Furthermore, we prove a superconvergence result, demonstrating that through appropriate post‐processing, one can achieve an enhanced convergence rate of in the ‐norm for sufficiently smooth solutions. The theoretical analysis employs a discrete fractional Grönwall inequality and rigorous error estimates that account for the weak singularity at the initial time. In contrast to existing methods such as Peng et al. (2024), which use first‐order L1 temporal discretization, our scheme achieves second‐order temporal accuracy and systematically handles variable coefficients and general distributed delay kernels. Numerical experiments validate the theoretical results and demonstrate the method's effectiveness for biologically relevant models, including a fractional Nicholson's blowflies equation with spatial heterogeneity. Additional experiments with higher‐order finite elements confirm the higher‐order spatial convergence, and condition number analyses demonstrate the mesh‐independent performance of the predictor‐corrector linearization. The robustness of the algorithm is confirmed through long‐time simulations with strongly variable coefficients and distributed delay kernels.