Exercise: Robust preconditioners

82. Exercise: Robust preconditioners#

Develop \(\varepsilon\)-robust preconditioners for the following problems:

82.1. Exercise A:#

\[ \Omega = (0,3) \times (0,1) \setminus (1,2) \times (\varepsilon , 1) \]
\[ \Gamma_D = \{ 0 \} \times (0,1) \]
\[ A(u,v) = \int \nabla u \nabla v \, dx \]

82.2. Exercise B:#

\[ \Omega_1 = (0,1) \times (0,1), \quad \Omega_2 = (1,2) \times (0,1) \]

Interface:

\[ \gamma = \overline{\Omega_1} \cap \overline{\Omega_2} \]
\[ V = H^1(\Omega_1) \times H^1(\Omega_2) \text{with Dirichlet at the bottomo} \]
\[ A(u,v) = \int_{\Omega_1 \cup \Omega_2} \nabla u \, \nabla v \, dx + \frac{1}{\varepsilon} \int_\gamma (u_1-u_2) (v_1 - v_2) \, ds \]