  Topology optimization problems generally support multiple local minima, and
real-world applications are typically three-dimensional. In previous work [I.
P. A. Papadopoulos, P. E. Farrell, and T. M. Surowiec, Computing multiple
solutions of topology optimization problems, SIAM Journal on Scientific
Computing, (2021)], the authors developed the deflated barrier method, an
algorithm that can systematically compute multiple solutions of topology
optimization problems. In this work we develop preconditioners for the linear
systems arising in the application of this method to Stokes flow, making it
practical for use in three dimensions. In particular, we develop a nested block
preconditioning approach which reduces the linear systems to solving two
symmetric positive-definite matrices and an augmented momentum block. An
augmented Lagrangian term is used to control the innermost Schur complement and
we apply a geometric multigrid method with a kernel-capturing relaxation method
for the augmented momentum block. We present multiple solutions in
three-dimensional examples computed using the proposed iterative solver.
