THEOREM 1 Suppose a mixed variational problem in abstract settingProof. See, e.g., .
Find such that
Assume that the following conditions are satisfied
(A1) , ,
(A2) and with
(A3) the LBB (Ladyzenskaya-Babuška-Brezzi) condition, i.e.
(A4) the - ellipticity, i.e. with
We apply Theorem 1 to the variational formulation (VF1).
THEOREM 2 Suppose that , and that is a piecewise constant function fulfillingProof. We have to verify the assumptions of Theorem 1.
(A1) It is the assumption, and .
(A2) Applying Lemma 3 we get
i.e., we can set , and
(A3) The LBB condition has been proved for the Stokes problem, cf. e.g. [6, Theorem 3.7]. There exists a constant with
(A4) We can prove ellipticity for the whole space , i.e.
We remark that it may be useful to choose depending on the actual coefficients , e.g., .