Uniform in time error estimates for a finite element method applied to a downscaling data assimilation algorithm for the Navier-Stokes equations

2018/07/23 by García-Archilla, Novo, Julia, Titi, Edriss S. · 3 citations
#35Q30 #65M12 #65M15 #65M20 #65M60 #65M70 #76B75 #Analysis of PDEs (math.AP) #Atmospheric and Oceanic Physics (physics.ao-ph) #FOS: Mathematics #FOS: Physical sciences #Fluid Dynamics (physics.flu-dyn) #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.1807.08735

Abstract

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.

Cited by

• Finite Element Analysis for the Chafee-Infante Equation Using Distributed Feedback Control
• Interpolated Discrepancy Data Assimilation for PDEs with Sparse Observations
• Data Assimilation in Large Eddy Simulation: Addressing Model-Observation Mismatch from Navier-Stokes Data

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