ement approximation of the Poisson equation on the unit square:</p>
$$- \nabla (d \nabla) u = f \; \hbox{in } (0,1)^2$$<p>for the following boundary conditions</p>
<ul>
<li>pure Dirichlet boundary condition: $u = g_D \text{ on } \partial \Omega$.</li>
<li>Pure Neumann boundary condition: $d\nabla u\cdot n=g_N \text{ on } \partial \Omega$.</li>
<li>mixed boundary condition: $u=g_D \text{ on }\Gamma_D, \nabla u\cdot n=g_N \text{ on }\Gamma_N.$</li>
</ul>
<p>Find $(\sigma , u)$ in $H_{g_N,\Gamma_N}(div,\Omega)