A Rational Approximation Scheme for Computing Mittag-Leffler Function with Discrete Elliptic Operator as Input

In this work, we propose a new scheme based on numerical quadrature to calculate the two-parameter Mittag-Leffler function with discrete elliptic operator $$-{\mathcal {L}}_h$$ as input. Except pure mathematical interest from approximation theory, our consideration also arises from solving sub-diffusion equations numerically with time-independent diffusion coefficient. We obtain the scheme by applying Gauss-Legendre quadrature rule for the integral representation of the Mittag-Leffler function. Rigorous error analysis is carried out which shows that the scheme converges exponentially with the increase of quadrature nodes. The computational cost of the algorithm is solving K sparse linear systems with K the number of quadrature nodes. It is worth to point out that the scheme is completely parallel which can save much time if the dimension of $${\mathcal {L}}_h$$ is very large. Some numerical tests are provided to verify the efficiency and robustness of our scheme.