I've coded up a Stokes Flow problem using finite elements and am in the process of verifying that it works. I'm just not sure what convergence rate I should be expecting as I globally refine the mesh.
I know for scalar problems using linear basis functions I'd expect order $h^2$ convergence ($h$ is element size), and using quadratic basis functions I'd expect order $h^3$ convergence in the $L^2$ norm and one power less in the $H^1$ seminorm. The problem I'm having now is that when coding Stokes flow I used the Taylor-Hood element which uses linears for the pressure and quadratic for the velocity components. Is it as simple as the velocities converging at $h^3$ and the pressure at order $h^2$?
I originally posted this on Mathoverflow, but now I feel that this might be a better forum for this question. There is actually a finite element tag here which is nice.
The standard a priori error estimate for the discrete solution $(u_h,p_h)$ of the Stokes problem with Taylor-Hood ($P_2-P_1$) elements is $$\|u-u_h\|_{1,\Omega} + \|p-p_h\|_{0,\Omega} \leq C h^2(|u|_{3,\Omega}+|p|_{2,\Omega}),$$ assuming that the solution $(u,p)$ is regular enough. It follows directly from the best approximation result and the interpolation properties of the approximation spaces.