Global in time stability and accuracy of IMEX-FEM data assimilation schemes for Navier–Stokes equations

Abstract

We study numerical schemes for incompressible Navier-Stokes equations using IMEX temporal discretizations, finite element spacial discretizations, and equipped with continuous data assimilation (a technique recently developed by Azouani, Olson, and Titi in 2014). We analyze stability and accuracy of the proposed methods, and are able to prove well-posedness, long time stability, and long time accuracy estimates, under restrictions of the time step size and data assimilation parameter. We give results for several numerical tests that illustrate the theory, and show that, for good results, the choice of discretization parameter and element choices can be critical. IntroductionData assimilation (DA) refers to a wide class of schemes for incorporating observational data in simulations, in order to increase the accuracy of solutions and to obtain better estimates of initial conditions. It is the subject of a large body of work (see, e.g., [13,31,33], and the references therein). DA algorithms are widely used in weather modeling, climate science, and hydrological and environmental forecasting [31]. Classically, these techniques are based on linear quadratic estimation, also known as the Kalman Filter. The Kalman Filter is described in detail in several textbooks, including [13,31,33,10], and the references therein.Recently, a promising new approach to data assimilation was pioneered by Azouani, Olson, and Titi [3,4] (see also [9,25,39] for early ideas in this direction). This new approach, which we call AOT Data Assimilation or continuous data assimilation, adds a feedback control term at the PDE level that nudges the computed solution towards the reference solution corresponding to the observed data. A similar approach is taken by Blömker, Law, Stuart, and Zygalakis in [7] in the context of stochastic differential equations. The AOT algorithm is based on feedback control at the PDE (partial differential equation) level, described below. The first works in this area assumed noise-free observations, but [5] adapted the method to the case of noisy data, and [19] adapted to the case in which measurements are obtained discretely in time and may be contaminated by systematic errors. Computational experiments on the AOT algorithm and its variants were carried * An important property of the b operator is that b(u, v, v) = 0 for u, v ∈ X.We will utilize the following bounds on b.Lemma 2.1. There exists a constant M > 0 dependent only on Ω satisfyingfor all u, v, w ∈ X for which the norms on the right hand sides are finite.Remark 2.2. Here and throughout, sharper estimates are possible if we restrict to 2D. However, for simplicity and generality, we do not make this restriction.Proof. These well known bounds follow from Hölder's inequality, Sobolev inequalities, and the Poincaré inequality. Discretization preliminariesDenote by τ h a regular, conforming triangulation of the domain Ω, and let X h ⊂ X, Q h ⊂ Q be an inf-sup stable pair of discrete velocity -pressure spaces. For simplicity, we will take X h = X ∩ P k and Q h = Q ∩...

Uniform in Time Error Estimates for a Finite Element Method Applied to a Downscaling Data Assimilation Algorithm for the Navier--Stokes Equations

“…More closely related to the present work are [34] and [38]. In [38] they only analyze linear problems and, for the proof of the results on the Navier-Stokes equations they present, they refer to [34] with some differences that they point out. They also present a wide collection of numerical experiments.…”

“…They also present a wide collection of numerical experiments. In [34], the authors consider fully discrete approximations to equation (2) where the spatial discretization is performed with a MFE Galerkin method plus graddiv stabilization. A second order IMEX in time scheme is analyzed in [34], and, as in [31], [37] and the present paper, uniform-in-time error bounds are obtained.…”

“…In [34], the authors consider fully discrete approximations to equation (2) where the spatial discretization is performed with a MFE Galerkin method plus graddiv stabilization. A second order IMEX in time scheme is analyzed in [34], and, as in [31], [37] and the present paper, uniform-in-time error bounds are obtained. Compared with [34], for the same convergence rate, the error bounds in the present paper have constants that do not depend on inverse powers of the viscosity parameter ν (Theorem 3.3) or, for similar error constants, error bounds in the present paper have an order of convergence one unit larger (Theorem 3.2 below).…”

“…A second order IMEX in time scheme is analyzed in [34], and, as in [31], [37] and the present paper, uniform-in-time error bounds are obtained. Compared with [34], for the same convergence rate, the error bounds in the present paper have constants that do not depend on inverse powers of the viscosity parameter ν (Theorem 3.3) or, for similar error constants, error bounds in the present paper have an order of convergence one unit larger (Theorem 3.2 below). Also, the analysis in [34] is restricted to I H u being an interpolant for non smooth functions (Clément, Scott-Zhang, etc), since it makes explicit use of bound (21), which is not valid for nodal (Lagrange) interpolation (neither it is (22)).…”

Continuous data assimilation reduced order models of fluid flow

“…Remark 3.7. If β 1 , β 2 are chosen to be 1/2, the condition α 1 > 0 reduces to ν − CµH 2 > 0, which is the same condition found in [23] and references therein, for a relationship between the nudging parameter, viscosity, and coarse mesh width. Choosing β 1 , β 2 larger can allow one to choose the coarse mesh width H larger (and thus require less observational data) while still satisfying α i > 0, i = 1, 2.…”