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

The mixed variational formulation

We establish a mixed variational formulation of the linear problem in the pair $\left( {\bf Y}, L^2_\star(\Omega) \right)$ of spaces. Let be

$$\begin{eqnarray*} {\bf Y}&=& H_0({ \bf rot };\Omega) \cap H(\div ;\Omega), \\... ... \quad \vert\vert.\vert\vert _{\bf M}= \vert\vert.\vert\vert _0 \end{eqnarray*}$$

where the space ${\bf Y}$ is equipped with the norm $\vert.\vert _{\bf Y}$, cf. (20). Using a trick similar to that in [1] we choose a constant $\rho = { \rm const }> 0$ and define

  

$$\hat{a}({\bf u},{\bf v}) = \int_{\Omega} \nu { \bf rot }{\bf u} \cdot { \bf rot }{\bf v} + \rho \div {\bf u} \div {\bf v} \;dx$$ (21)

$$b({\bf v},\lambda) = - \int_{\Omega} \lambda \div {\bf v} \; dx.$$ (22)

Then we may write the standard variational formulation:
(VF1) Find a pair $({\bf u},\lambda) \in {\bf Y}\times L^2_\star(\Omega)$ such that

   

$$\hat{a}({\bf u},{\bf v}) + b({\bf v}, \lambda) = \langle {\bf f},{\bf v} \rangle \quad \forall {\bf v} \in {\bf Y},$$ (23)

$$b({\bf u}, \mu) = 0 \quad \forall \mu \in {\bf M}.$$ (24)

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