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Finite element methods based on two families of second-order numerical formulas for the fractional Cable model with smooth solutions

We apply two families of novel fractional $θ$-methods, the FBT-$θ$ and FBN-$θ$ methods developed by the authors in previous work, to the fractional Cable model, in which the time direction is approximated by the fractional $θ$-methods, and the space direction is approximated by the finite element method. Some positivity properties of the coefficients for both of these methods are derived, which are crucial for the proof of the stability estimates. We analyse the stability of the scheme and derive an optimal convergence result with $O(τ^2+h^{r+1})$ for smooth solutions, where $τ$ is the time mesh size and $h$ is the spatial mesh size. Some numerical experiments with smooth and nonsmooth solutions are conducted to confirm our theoretical analysis. To overcome the singularity at initial value, the starting part is added to restore the second-order convergence rate in time.