Pointwise-in-Time Error Analysis of the Nonuniform Alikhanov Scheme for the Fractional Wave Equation

This paper develops a fully discrete finite element scheme for the fractional wave equation with order α∈(1,2), whose solution typically exhibits a weak singularity near the initial time t=0. By introducing an auxiliary variable, we first reformulate the fractional wave problem into an equivalent coupled system of two fractional equations. The resulting coupled system is then discretized using the nonuniform Alikhanov formula in time and the standard finite element method on triangular meshes in space. Through rigorous analysis, we establish a pointwise-in-time error estimate for the proposed scheme in the H1 semi-norm. A key advantage of the proposed methodology is its ability to employ a sparser mesh near the initial time to achieve optimal convergence of local errors. In particular, our analysis shows that away from the initial time, the local rate of convergence reaches O(N−2) in time for r≥2. Finally, numerical experiments are given to verify the sharpness of the theoretical convergence rates.