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Simple and efficient continuous data assimilation of evolution equations via algebraic nudging
Author(s) -
Rebholz Leo G.,
Zerfas Camille
Publication year - 2021
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.22751
Subject(s) - discretization , mathematics , piecewise , diagonal , operator (biology) , finite element method , interpolation (computer graphics) , algebraic number , simple (philosophy) , projection (relational algebra) , constant (computer programming) , linear interpolation , mathematical analysis , algorithm , computer science , geometry , physics , philosophy , epistemology , animation , biochemistry , chemistry , computer graphics (images) , repressor , transcription factor , polynomial , gene , programming language , thermodynamics
Abstract We introduce, analyze, and test an interpolation operator designed for use with continuous data assimilation (DA) of evolution equations that are discretized spatially with the finite element method. The interpolant is constructed as an approximation of the L 2 projection operator onto piecewise constant functions on a coarse mesh, but which allows nudging to be done completely at the linear algebraic level, independent of the rest of the discretization, with a diagonal matrix that is simple to construct; it can even completely remove the need for explicit construction of a coarse mesh. We prove the interpolation operator has sufficient stability and accuracy properties, and we apply it to algorithms for both fluid transport DA and incompressible Navier–Stokes DA. For both applications we prove the DA solutions with arbitrary initial conditions converge to the true solution (up to discretization error) exponentially fast in time, and are thus long‐time accurate. Results of several numerical tests are given, which both illustrate the theory and demonstrate its usefulness on practical problems.