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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!

yemino
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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).

knl
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  • do you know some stabilized method P1-P0? I'm looking for just for practice the programming – yemino Apr 29 '17 at 19:10
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    I'm sorry, I do not know of any $P_1$-$P_0$ stabilized methods for Stokes. The standard (consistent) residual stabilization works for any continuous pressure. For discontinous pressure you need higher-order velocity space (i.e. $P_2$-$P_0$ in two-dimensional problems), see e.g. Stenberg, Videman - On the error analysis of stabilized finite element methods for the Stokes problem. You can find the preprint in arXiv. – knl Apr 29 '17 at 20:21