TY - JOUR
T1 - Mesh-Free Interpolant Observables for Continuous Data Assimilation
AU - Biswas , Animikh
AU - Brown , Kenneth R.
AU - Martinez , Vincent R.
JO - Annals of Applied Mathematics
VL - 3
SP - 296
EP - 355
PY - 2022
DA - 2022/08
SN - 38
DO - http://doi.org/10.4208/aam.OA-2022-0006
UR - https://global-sci.org/intro/article_detail/aam/20880.html
KW - Continuous data assimilation, nudging, 2D Navier-Stokes equations, general interpolant observables, synchronization, higher-order convergence, partition of unity, mesh-free, Azounai-Olson-Titi algorithm.
AB - <p style="text-align: justify;">This paper is dedicated to the expansion of the framework of general
interpolant observables introduced by Azouani, Olson, and Titi for continuous
data assimilation of nonlinear partial differential equations. The main feature of
this expanded framework is its mesh-free aspect, which allows the observational
data itself to dictate the subdivision of the domain via partition of unity in
the spirit of the so-called Partition of Unity Method by Babuska and Melenk.
As an application of this framework, we consider a nudging-based scheme for
data assimilation applied to the context of the two-dimensional Navier-Stokes
equations as a paradigmatic example and establish convergence to the reference
solution in all higher-order Sobolev topologies in a periodic, mean-free setting.
The convergence analysis also makes use of absorbing ball bounds in higher-order Sobolev norms, for which explicit bounds appear to be available in the
literature only up to $H^2;$ such bounds are additionally proved for all integer
levels of Sobolev regularity above $H^2.$</p>