The paper is concerned with computable estimates
of the distance between a vector-valued function in
the Sobolev space (where and
Ω is a bounded Lipschitz domain in ℝd)
and the subspace containing all divergence-free (solenoidal) vector functions. Derivation of these estimates is closely related to the stability theorem that establishes
existence of a bounded operator inverse to the operator .
The constant in the respective stability inequality
arises in the estimates of the distance to the set . In general,
it is difficult to find a guaranteed and realistic upper bound of this global constant.
We suggest a way to circumvent this difficulty by using
weak (integral mean) solenoidality conditions and
localized versions of the stability theorem.
They are derived for the case where
Ω is represented
as a union of simple subdomains (overlapping or non-overlapping),
for which estimates of the corresponding stability constants are known.
These new versions of the stability theorem
imply estimates of the distance to that involve only
local constants associated with subdomains. Finally, the estimates are used for deriving fully computable
a posteriori estimates for problems in the theory of incompressible viscous fluids.