  We propose a Nested BDDC for a class of saddle-point problems. The method
solves for both flux and pressure variables. The fluxes are resolved in
three-steps: the coarse solve is followed by subdomain solves, and last we look
for a divergence-free flux correction and pressure variables using conjugate
gradients with a Multilevel BDDC preconditioner. Because the coarse solve in
the first step has the same structure as the original problem, we can use this
procedure recursively and solve (a hierarchy of) coarse problems only
approximately, utilizing the coarse problems known from the BDDC. The resulting
algorithm thus first performs several upscaling steps, and then solves a
hierarchy of problems that have the same structure but increase in size while
sweeping down the levels, using the same components in the first and in the
third step on each level, and also reusing the components from the higher
levels. Because the coarsening can be quite aggressive, the number of levels
can be kept small and the additional computational cost is significantly
reduced due to the reuse of the components. We also provide the condition
number bound and numerical experiments confirming the theory.
