68. Exercise: Robust preconditioners#
Develop \(\varepsilon\)-robust preconditioners for the following problems:
68.1. Exercise A:#
\[
\Omega = (0,3) \times (0,1) \setminus (1,2) \times (\varepsilon , 1)
\]
\[
\Gamma_D = \{ 0 \} \times (0,1)
\]
\[
A(u,v) = \int \nabla u \nabla v \, dx
\]
68.2. Exercise B:#
\[
\Omega_1 = (0,1) \times (0,1), \quad \Omega_2 = (1,2) \times (0,1)
\]
Interface:
\[
\gamma = \overline{\Omega_1} \cap \overline{\Omega_2}
\]
\[
V = H^1(\Omega_1) \times H^1(\Omega_2) \text{with Dirichlet at the bottomo}
\]
\[
A(u,v) = \int_{\Omega_1 \cup \Omega_2} \nabla u \, \nabla v \, dx
+ \frac{1}{\varepsilon} \int_\gamma (u_1-u_2) (v_1 - v_2) \, ds
\]