Unconditional Optimal Error Estimates of a Linearized Nonuniform Alikhanov Scheme for Nonlinear Superdiffusion Equations

In this paper, we consider a nonlinear fractional superdiffusion equation involving a Caputo derivative of order α∈(1,2). Solutions to this class of problems typically exhibit weak singularities near the initial time. To handle this singularity, we introduce an auxiliary variable p:=Dtα/2(u−tu1) and reformulate the original problem as an equivalent coupled system. This system is then discretized using the nonuniform Alikhanov scheme in time, the conforming Galerkin finite element method in space, and Newton linearization for the nonlinear term. We prove that the numerical solutions remain uniformly bounded, independent of the temporal and spatial mesh parameters. Combining this result with a discrete fractional Grönwall inequality and a temporal-spatial splitting argument, we establish unconditional optimal error estimates of order O(N−min{2,rα/2}+hk). Finally, numerical experiments are presented to verify the sharpness of the theoretical convergence rates.