I'm looking for some algorithm of a finite elements to solve the problem
Find $u,p\in H_0^1(\Omega)\times L^2(\Omega)$ such that
$-\nu\Delta u+\nabla p=f$ in $\Omega\subset\mathbb{R}^2$
$\nabla\cdot u=0$ in $\Omega$
$u=0$ in $\partial\Omega$
where $\Omega$ is a bounded subset o $\mathbb{R}^2$, and we have a typical finite element mesh of $\Omega$ composed by triangles (its union is all $\Omega$, are not overlaped...), that is, the simplest and classic Stokes equations with lowest finite elements, nothings strange here.
I like to program a finite element code (just for practice) where the velocity $u_h\in H_h$ (continuous over $\Omega$) be approximate by piecewise linear elements over triangles and $p_h\in Q_h$ constants over each triangle of the mesh. $u_h$ and $p_h$ (discontinuous over $\Omega$) are the approximations by the finite elements method.
Do you know some easy finite method for practice? Some book, paper, pdf... with friendy notation.
I know that the scheme: find $(u_h,p)\in H_h\times Q_h$
$(\nu\nabla u,\nabla v)-(\nabla\nabla u,q)+(\nabla\nabla v,p)=(f,v)$ for all $v\in H_h$ and $q\in Q$
is not well possed. For that reason I'm looking for some well-posed scheme for practice the programming of the method.
