# Imposing Dirichlet Boundary Conditions with Lagrange Multipliers: stable mixed formulation

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?

• There are some old articles from the 70's by Ivo Babuška where they study methods like this. The main point is that you cannot choose the discrete spaces arbitrarily. If you have a code that supports it, I suggest that you decrease the order of the Lagrange multiplier space to e.g. $P_0$ discontinuos. I cannot remember what spaces work but usually decreasing the order of Lagrange multipliers is the right direction. – knl Sep 7 '17 at 6:10
• Hi, thank you for the suggestion. Actually the scheme P1/P1 works. I did some mistake on the right side. Anyway, the Babuska's work is pretty important for the scheme i got. Right now I do not implement the DG formulation. – Petr Henyš Sep 12 '17 at 11:39