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Journal article
Fully discrete numerical schemes of a data assimilation algorithm: uniform-in-time error estimates
IMA JOURNAL OF NUMERICAL ANALYSIS, v 40(4), pp 2584-2625
Oct 2020
Abstract
Our aim is to approximate a reference velocity field solving the two-dimensional Navier-Stokes equations (NSE) in the absence of its initial condition by utilizing spatially discrete measurements of that field, available at a coarse scale, and continuous in time. The approximation is obtained via numerically discretizing a downscaling data assimilation algorithm. Time discretization is based on semiimplicit and fully implicit Euler schemes, while spatial discretization (which can be done at an arbitrary scale regardless of the spatial resolution of the measurements) is based on a spectral Galerkin method. The two fully discrete algorithms are shown to be unconditionally stable, with respect to the size of the time step, the number of time steps and the number of Galerkin modes. Moreover, explicit, uniformin-time error estimates between the approximation and the reference solution are obtained, in both the L-2 and H-1 norms. Notably, the two-dimensional NSE, subject to the no-slip Dirichlet or periodic boundary conditions, are used in this work as a paradigm. The complete analysis that is presented here can be extended to other two- and three-dimensional dissipative systems under the assumption of global existence and uniqueness.
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Details
- Title
- Fully discrete numerical schemes of a data assimilation algorithm: uniform-in-time error estimates
- Publication Details
- IMA JOURNAL OF NUMERICAL ANALYSIS, v 40(4), pp 2584-2625
- Publisher
- OXFORD UNIV PRESS; OXFORD
- Number of pages
- 0
- Grant note
- National Science Foundation (NSF) (DMS-1516866); Office of Naval Research (ONR) (N00014-151-2333); Einstein Stiftung/Foundation - Berlin, Einstein Visiting Fellow Program (to E.S.T.); Ecole Polytechnique Foundation for its partial financial support through the 2017-2018 `Gaspard Monge Visiting Professor' Program (to E.S.T.).
- Resource Type
- Journal article
- Language
- English
- Academic Unit
- Drexel University
- Web of Science ID
- WOS:000610489200015
- Scopus ID
- 2-s2.0-85101010273
- Other Identifier
- 991021860675604721
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- Collaboration types
- Domestic collaboration
- International collaboration
- Web of Science research areas
- Mathematics, Applied