Error analysis of proper orthogonal decomposition data assimilation schemes with grad–div stabilization for the Navier–Stokes equations

• Published in: Journal of computational and applied mathematics Vol. 411; p. 114246
• Main Authors: García-Archilla, Bosco, Novo, Julia, Rubino, Samuele
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.09.2022
• Subjects: Data assimilation; Fully discrete schemes; Mixed finite elements methods; Navie–Stokes equations; Proper orthogonal decomposition; Uniform-in-time error estimates; 65M15; 76B75; Proper orthogonal decomposition; Fully discrete schemes; 35Q30; Uniform-in-time error estimates; Navie–Stokes equations; 65M70; 65M60; 65M12; Mixed finite elements methods; 65M20; Data assimilation
• ISSN: 0377-0427; 1879-1778; 1879-1778
• DOI: 10.1016/j.cam.2022.114246

The error analysis of a proper orthogonal decomposition (POD) data assimilation (DA) scheme for the Navier–Stokes equations is carried out. A grad–div stabilization term is added to the formulation of the POD method. Error bounds with constants independent on inverse powers of the viscosity parameter are derived for the POD algorithm. No upper bounds in the nudging parameter of the data assimilation method are required. Numerical experiments show that, for large values of the nudging parameter, the proposed method rapidly converges to the real solution, and greatly improves the overall accuracy of standard POD schemes up to low viscosities over predictive time intervals.