  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.
