Uniform in time error estimates for a finite element method applied to a downscaling data assimilation algorithm for the Navier-Stokes equations
Provides rigorous error analysis for data assimilation in fluid dynamics, benefiting numerical weather prediction and ocean modeling.
The paper proves uniform-in-time error estimates for a finite element method applied to a downscaling data assimilation algorithm for the Navier-Stokes equations, achieving error bounds independent of inverse powers of viscosity with grad-div stabilization. Numerical experiments confirm the theory.
In this paper we analyze a finite element method applied to a continuous downscaling data assimilation algorithm for the numerical approximation of the two and three dimensional Navier-Stokes equations corresponding to given measurements on a coarse spatial scale. For representing the coarse mesh measurements we consider different types of interpolation operators including a Lagrange interpolant. We obtain uniform-in-time estimates for the error between a finite element approximation and the reference solution corresponding to the coarse mesh measurements. We consider both the case of a plain Galerkin method and a Galerkin method with grad-div stabilization. For the stabilized method we prove error bounds in which the constants do not depend on inverse powers of the viscosity. Some numerical experiments illustrate the theoretical results.