  A stationary Stokes problem with a piecewise constant viscosity coefficient
in multiple subdomains is considered in the paper. For standard finite element
pairs, a robust inf-sup condition is required to show the robustness of the
discretization error with respect to the discontinuous viscosity, which has
only been proven for the two-subdomain case in the paper [Numer. Math. (2006)
103: 129--149]. To avoid the robust inf-sup condition of a discrete finite
element pair for multiple subdomains, we propose an ultra-weak augmented mixed
finite element formulation. By adopting a Galerkin-least-squares method, the
augmented mixed formulation can achieve stability without relying on the
inf-sup condition in both continuous and discrete settings. The key step to
having a robust priori error estimate is to use two norms, one energy norm and
one full norm, in robust continuity. The robust coercivity is proved for the
energy norm. A robust a priori error estimate in the energy norm is then
derived with the best approximation property in the full norm for the case of
multiple subdomains. Additionally, the paper introduces a singular Kellogg-type
example with exact solutions for the first time. Extensive numerical tests are
conducted to validate the robust error estimate.
