I'm looking for a book or pdf to study the Stokes problem with finite elements method
$\Delta u+\nabla p=f$ in $\Omega$
$\nabla\cdot u=0$ in $\Omega$
$+$ boundary conditions (example: $u=0$ on $\partial\Omega$.
I'm interested in study the existence and uniqueness of the continuous and discrete problem, in particular the $\mathbb{P}_1/\mathbb{P}_0$ formulation ($\mathbb{P}_1$ for $u$ and $\mathbb{P}_0$ for $p$).
Thanks!
The standard reference is Mixed Finite Element Methods and Applications by Boffi-Brezzi-Fortin. The combination $P_1-P_0$ will not, in general, satisfy the discrete inf-sup condition and therefore you must either stabilize the problem or use the so-called nonconforming $P_1-P_0$ approximation where the velocity is continuous only at the triangle edge midpoints (i.e., Crouzeix-Raviart shape functions).