  In this article, we design and analyze an arbitrary-order stabilized finite
element method to approximate the unique continuation problem for laminar
steady flow described by the linearized incompressible Navier--Stokes equation.
We derive quantitative local error estimates for the velocity, which account
for noise level and polynomial degree, using the stability of the continuous
problem in the form of a conditional stability estimate. Numerical examples
illustrate the performances of the method with respect to the polynomial order
and perturbations in the data. We observe that the higher order polynomials may
be efficient for ill-posed problems, but are also more sensitive for problems
with poor stability due to the ill-conditioning of the system.
