  We introduce an adaptive element-based domain decomposition (DD) method for
solving saddle point problems defined as a block two by two matrix. The
algorithm does not require any knowledge of the constrained space. We assume
that all sub matrices are sparse and that the diagonal blocks are spectrally
equivalent to a sum of positive semi definite matrices. The latter assumption
enables the design of adaptive coarse space for DD methods that extends the
GenEO theory to saddle point problems. Numerical results on three dimensional
elasticity problems for steel-rubber structures discretized by a finite element
with continuous pressure are shown for up to one billion degrees of freedom.
