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

TL;DR

This paper develops a high-order stabilized finite element method for the linearized Navier-Stokes equations to improve approximation of the unique continuation problem, analyzing error estimates and numerical performance.

Contribution

It introduces an arbitrary-order stabilized finite element approach with error analysis tailored for the linearized Navier-Stokes equations in ill-posed problems.

Findings

Higher order polynomials can be efficient for ill-posed problems.

The method's performance depends on polynomial degree and data perturbations.

Numerical results confirm the theoretical error estimates.

Abstract

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.