2024/01/12 by Bosco Garcia-Archilla, Xuejian Li, Garcia-Archilla, Bosco +5 · 1 citation
Engineering · #65N12 #65N30 #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Numerical Analysis (math.NA) #Reservoir Engineering and Simulation Methods #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2401.06749
openalex publication_date 2024/01/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider nonlinear solvers for the incompressible, steady (or at a fixed time step for unsteady) Navier-Stokes equations in the setting where partial measurement data of the solution is available. The measurement data is incorporated/assimilated into the solution through a nudging term addition to the the Picard iteration that penalized the difference between the coarse mesh interpolants of the true solution and solver solution, analogous to how continuous data assimilation (CDA) is implemented for time dependent PDEs. This was considered in the paper [Li et al. \it CMAME 2023], and we extend the methodology by improving the analysis to be in the L2 norm instead of a weighted H1 norm where the weight depended on the coarse mesh width, and to the case of noisy measurement data. For noisy measurement data, we prove that the CDA-Picard method is stable and convergent, up to the size of the noise. Numerical tests illustrate the results, and show that a very good strategy when using noisy data is to use CDA-Picard to generate an initial guess for the classical Newton iteration.