When developing robust preconditioners for multiphysics problems, fractional functions of the Laplace operator often arise and need to be inverted. Rational approximation in the uniform norm can be used to convert inverting those fractional operators into inverting a series of shifted Laplace operators. Care must be taken in the approximation so that the shifted Laplace operators remain symmetric positive definite, making them better conditioned. In this work, we study two greedy algorithms for finding rational approximations to such fractional operators. The first algorithm improves the orthogonal greedy algorithm discussed in [Li et al., SISC, 2024] by adding one minimization step in the uniform norm to the procedure. The second approach employs the weak Chebyshev greedy algorithm in the uniform norm. Both methods yield non-increasing error. Numerical results confirm the effectiveness of our proposed algorithms, which are also flexible and applicable to other approximation problems. Moreover, with effective rational approximations to the fractional operator, the resulting algorithms show good performance in preconditioning a Darcy-Stokes coupled problem.

翻译：在开发多物理场问题的鲁棒预处理器时，经常出现拉普拉斯算子的分数阶函数并需要求逆。利用一致范数下的有理逼近，可将这些分数阶算子的求逆转化为一系列平移拉普拉斯算子的求逆。逼近过程中需谨慎处理，以确保平移后的拉普拉斯算子保持对称正定，从而改善其条件数。本文研究了两种用于求解此类分数阶算子有理逼近的贪心算法。第一种算法改进了[Li et al., SISC, 2024]中讨论的正交贪心算法，在流程中增加了一致范数下的最小化步骤。第二种方法采用了一致范数下的弱切比雪夫贪心算法。两种方法均能产生非递增误差。数值结果验证了所提算法的有效性，这些算法也具备灵活性，可适用于其他逼近问题。此外，通过对分数阶算子进行有效的有理逼近，所得算法在达西-斯托克斯耦合问题的预处理中表现出良好的性能。