Why spaces of order 2 and 1 to solve the Navier-Stokes equations?

Dear all,
I see that in order to solve the Navier Stokes equations with Fenics, see for example here,

a function space of order 2 is used for the velocity and one of order 1 for the pressure

V = VectorFunctionSpace(mesh, 'P', 2)
Q = FunctionSpace(mesh, 'P', 1)

Changing the order of these spaces may make the algorithm yield the wrong solution. May you please give me some references explaining why these orders 2+1 are chosen?

Thank you

See: The Stokes equations — FEniCS 22 tutorial
and in general Ladyzhenskaya–Babuška–Brezzi condition - Wikipedia
A general overview is given in: https://www.researchgate.net/publication/222568702_Numerical_methods_for_incompressible_viscous_flow
See section surrounding equation 26.

This is a question that is discussed in any CFD course you may get offered at your university, as long as that course discusses FEM.

We are happy to help with these and other questions, but we definitely can’t replace your lecturer.

A further suggestion of a very comprehensive book on the topic is Finite Element Methods for Incompressible Flow Problems | SpringerLink . I personally use that book for the CFD course I am delivering at MSc level for math students.

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