HEJ, HU ISSN 1418-7108
Manuscript no.: ANM-981030-A
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The Brezzi theorem

We recall the Brezzi theorem, e.g. from [3].

THEOREM 1   Suppose a mixed variational problem in abstract setting
Find $({\bf u},\lambda) \in {\bf X}\times {\bf M}$ such that
$\displaystyle {a}({\bf u},{\bf v}) + b({\bf v}, \lambda)$ $\textstyle =$ $\displaystyle \langle {\bf f},{\bf v} \rangle
\quad \forall {\bf v} \in {\bf X},$ (25)
$\displaystyle b({\bf u}, \mu)$ $\textstyle =$ $\displaystyle \langle g, \mu \rangle \quad \forall \mu \in {\bf M}.$ (26)

Assume that the following conditions are satisfied
(A1) ${\bf f} \in {\bf X}^{\star}$, $g \in {\bf M}^{\star}$,
(A2) $\exists \alpha_2 = { \rm const }> 0$ and $\beta_2 = { \rm const }> 0$ with

\vert a({\bf u},{\bf v})\vert &\leq& \alpha_2 \vert\vert{\bf ...
...f M}
\quad \forall {\bf v} \in {\bf X}, \forall \mu\in{\bf M},

(A3) the LBB (Ladyzenskaya-Babuška-Brezzi) condition, i.e.
$\exists \beta_1 = { \rm const }> 0$ such that
\sup_{{\bf v}\in{\bf X}, {\bf v}\neq{\bf0}}
\frac{b({\bf ...
...ert\vert\mu\vert\vert _{\bf M}\quad
\forall \mu \in {\bf M},
\end{displaymath} (27)

(A4) the ${{\bf V}_0}$ - ellipticity, i.e. $\exists \alpha_1 = { \rm const }> 0$ with
a({\bf v},{\bf v}) \geq \alpha_1 \vert\vert{\bf v}\vert\vert^2_{\bf X}
\quad \forall {\bf v} \in {{\bf V}_0}
\end{displaymath} (28)

{{\bf V}_0}= {\bf V}(0) = \left\{ {\bf v} \in {\bf X}: b({\bf v},\mu) = 0 \quad
\forall \mu \in {\bf M}\right\}.
\end{displaymath} (29)

Then there exists a unique solution $({\bf u},\lambda) \in {\bf X}\times {\bf M}$ and the a-priori estimates

\vert\vert{\bf u}\vert\vert _{\bf X}&\leq& \frac{1}{\alpha_1}...
...{\alpha_1} \right)
\vert\vert g\vert\vert _{{\bf M}^{\star}}


Proof. See, e.g., [4].
We apply Theorem 1 to the variational formulation (VF1).

THEOREM 2   Suppose that ${\bf f}\in{\bf Y}^{\star}$, $\rho = { \rm const }> 0$ and that $\nu(x)$ is a piecewise constant function fulfilling
0 < \underline{\nu} \leq \nu(x) \leq \bar{\nu}.
\end{displaymath} (30)

Then (23), (24) has a unique solution $({\bf u},\lambda) \in
{\bf Y}\times L^2_\star(\Omega)$ for which a-priori estimates

\vert{\bf u}\vert _1 &\leq& \frac{1}{c_{\rm L}}  \vert\vert{...
...t)}{c_{\rm L}})
\vert\vert{\bf f}\vert\vert _{{\bf Y}^{\star}}


Proof. We have to verify the assumptions of Theorem 1.
(A1) It is the assumption, and $g=0$.
(A2) Applying Lemma 3 we get

\left\vert \hat{a}({\bf u},{\bf v}) \right\vert &=&
...t) \vert{\bf u}\vert _{\bf Y}\; \vert{\bf v}\vert _{\bf Y}, \\

i.e., we can set $\alpha_2 = \max \left( \bar{\nu}, \rho \right) $, and

\left\vert b({\bf v},\mu) \right\vert &=&
\left\vert \int_{...
...\leq& \vert\vert\mu\vert\vert _0 \; \vert{\bf v}\vert _{\bf Y},

i.e. $\beta_2 = 1$.
(A3) The LBB condition has been proved for the Stokes problem, cf. e.g. [6, Theorem 3.7]. There exists a constant $c_{\rm L} > 0$ with
\sup_{{\bf v}\in H_0^1(\Omega)^3, {\bf v}\neq{\bf0}}
...ert\mu\vert\vert _0 \quad
\forall \mu \in L^2_\star(\Omega).
\end{displaymath} (31)

>From Lemma 1 and (20) we get $\vert{\bf v}\vert _{\bf Y}= \vert{\bf v}\vert _1 \;\forall {\bf v}\in H_0^1(\Omega)^3$, and with $H_0^1(\Omega)^3 \subset {\bf Y}$ it follows that
\sup_{{\bf v}\in {\bf Y}, {\bf v}\neq{\bf0}}
\frac{ \int_...
...ert\mu\vert\vert _0 \quad
\forall \mu \in L^2_\star(\Omega).
\end{displaymath} (32)

(A4) We can prove ellipticity for the whole space ${\bf Y}$, i.e.
$\displaystyle \hat{a}({\bf v},{\bf v})$ $\textstyle \geq$ $\displaystyle \underline{\nu} \vert\vert{ \bf rot }{\bf v}\vert\vert _0^2
+ \rho \vert\vert\div {\bf v}\vert\vert _0^2 \quad \forall {\bf v} \in {\bf Y}$  
$\displaystyle \hat{a}({\bf v},{\bf v})$ $\textstyle \geq$ $\displaystyle \min \left( \underline{\nu}, \rho \right) \vert{\bf v}\vert _{\bf Y}^2
\quad \forall {\bf v} \in {\bf Y},$ (33)

and $\alpha_1 = \min \left( \underline{\nu}, \rho \right)$.
We remark that it may be useful to choose $\rho$ depending on the actual coefficients $\nu$, e.g., $\underline\nu\leq \rho \leq\bar\nu$.
HEJ, HU ISSN 1418-7108
Manuscript no.: ANM-981030-A
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