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.

在这项工作中，我们提出了一种基于数值正交的新方案，以离散椭圆运算符\（-{\ mathcal {L}} _ h \）作为输入来计算两参数Mittag-Leffler函数。除了来自逼近理论的纯粹数学兴趣外，我们的考虑还来自于求解具有时间独立扩散系数的子扩散方程。我们通过对高斯-莱格勒函数的积分表示应用高斯-勒根德勒正交规则来获得该方案。进行了严格的误差分析，结果表明该方案随着正交节点的增加而呈指数收敛。该算法的计算成本是用K求解K个稀疏线性系统正交节点数。值得指出的是，该方案是完全并行的，如果\（{\ mathcal {L}} _ h \）的尺寸很大，可以节省大量时间。提供了一些数值测试，以验证我们的方案的效率和鲁棒性。