What would be the optimal convergence rate for the Stokes problem?

General
1

Hello everyone,

Currently, I’m addressing the Stokes problem and observing a convergence rate of around 3 in the (L2 norm) for velocity. Considering I’m employing P2 for velocity and P1 for pressure, what would be the optimal convergence rate for the Stokes problem? Shouldn’t I expect a convergence rate closer to 2, given the use of P2 for velocity?

Thank you in advance.

2

Given a Stokes problem posed with sufficiently smooth analytical solution and material cofficients discrectised on a mesh of granularity measured to be h, the standard Taylor-Hood element of degree \ell in the velocity approximation and \ell-1 in the pressure approximation would yield expected approximation error convergence rates of:

  • \Vert \vec{u} - \vec{u}_h \Vert_{L_2} \backsim \mathcal{O(h^{\ell+1})}
  • \Vert \vec{u} - \vec{u}_h \Vert_{H^1} \backsim \mathcal{O(h^{\ell})}
  • \Vert p - p_h \Vert_{L_2} \backsim \mathcal{O(h^{\ell})}
  • \Vert p - p_h \Vert_{H^1} \backsim \mathcal{O(h^{\ell-1})}
3

Thank you for your response. Do you have any articles or resources that can provide detailed explanations regarding why the convergence rates should be close to 2 or 3, as you mentioned?

4

Maybe start with chapter 5 of: https://www.ljll.fr/pironneau/publi/publications/OPfemInFluids.pdf
Referring to chapter 3 of:
https://webs.um.es/eliseo/um/uploads/Main/Girault_Raviart_Finite_Element_Approximation_of_the_Navier_Stokes_Equations_LNM_1979.pdf

Allthough it should be in more modern textbooks as well

5

As @dokken has already mentioned, this result derives from standard a priori error analysis. You should be able to find it in any standard introductory FEM text, or lecture notes.

6

Thank you, sir, for the answer.

7

OK, think you so much for you answer.