A corrected Alikhanov scheme for a subdiffusion equation

Abstract

This paper considers the subdiffusion equation with a weakly singular solution. To achieve the optimal accuracy with a smaller grading parameter r than that of the standard Alikhanov scheme, it’s essential to correct the Alikhanov scheme and the approximation $v^{n-\theta }\approx \theta v^n+(1-\theta )v^{n-1}$ (0 < θ < 1) by investigating the corrected terms ${\beta }_n^{\sigma }(v^1-v^0)$ and ${\mu }_n^{\sigma }(v^1-v^0)$, respectively. After that, the truncation error of the corrected Alikhanov scheme and the corrected approximation of $v^{n-\theta }$ are given. By adopting the corrected Alikhanov scheme for the Caputo derivative and the corrected approximation for $v^{n-\theta }$, we construct a fully discrete scheme for the subdiffusion equation, employing a standard finite element method in space. Furthermore, the stability analysis and the optimal convergent analysis for the proposed scheme are investigated. Finally, numerical experiments are conducted to verify the theoretical results.