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

Data assimilation finite element method for the linearized Navier-Stokes equations with higher order polynomial approximation