I have a simple elastostatic problem defined by the following weak formulation:
Find $u$ for $\forall v \in H^{1}$
$\int_{\Omega}\epsilon(u):\sigma(v)\ d\Omega = 0$
$u = u_{0}\ on\ \Gamma_{D}$
Where $\epsilon$ and $\sigma$ are strain and stress tensor and the $u$ is a displacement vector. Now introduce Lagrange multipliers $\lambda$ to impose boundary conditions on $\Gamma_{D}$. New mixed formulation i formulated looks:
Find $u,\lambda$:
$\int_{\Omega}\epsilon(u):\sigma(v)\ d\Omega + \int_{\Gamma_{D}}\lambda\cdot v\ ds + \int_{\Gamma_{D}}\mu\cdot u\ ds$$= \int_{\Gamma_{D}}\mu\cdot u_{0}\ ds$
Where $v$ and $\mu$ are test functions from a mixed space $H^{1}\times H^{1}$ (linear P1/P1).
The formulation seems not work. Could you give me some hints how to quickly investigate the properties of defined mixed scheme?