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| Mesh-Free Interpolant Observables forContinuous Data Assimilation |
| Animikh Biswas,Kenneth R. Brown,Vincent R. Martinez |
| (Department of Mathematics & Statistics, University of Maryland–Baltimore
County, 1000 Hilltop Circle, Baltimore, MD 21250, USA;Department of Mathematics, University of California–Davis, One Shields
Avenue, Davis, CA 95616, USA;Department of Mathematics & Statistics, CUNY Hunter College, 695
Park Ave, New York, NY 10065, USA;
Department of Mathematics, CUNY Graduate Center, 365 5th Ave, New
York, NY 10016, USA) |
| DOI： |
| Abstract: |
| 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 higherorder Sobolev norms, for which explicit bounds appear to be available in the
literature only up to H2
; such bounds are additionally proved for all integer
levels of Sobolev regularity above H2
. |
| Key words: Continuous data assimilation, nudging, 2D Navier-Stokes equations, general interpolant observables, synchronization, higher-order convergence, partition of unity,
mesh-free, Azounai-Olson-Titi algorithm. |