Abstract
In this paper, we are interested in designing and analyzing a finite element data assimilation method for laminar steady flow described by the linearized incompressible Navier–Stokes equation. We propose a weakly consistent stabilized finite element method which reconstructs the whole fluid flow from noisy velocity measurements in a subset of the computational domain. Using the stability of the continuous problem in the form of a three balls inequality, we derive quantitative local error estimates for the velocity. Numerical simulations illustrate these convergence properties and we finally apply our method to the flow reconstruction in a blood vessel.
Highlights
The question of how to assimilate measured data into large scale computations of flow problems is receiving increasing attention from the computational community
One situation is when the data necessary to make the flow problem well posed is lacking, that is, boundary data can not be obtained on parts of the boundary, but some other measured data on the boundary or in the bulk is available to make up for this shortfall, or alternatively the position of the boundary itself is unknown
Other examples can be found in [25], where additional measured data is used to compensate for a lack of knowledge of the boundary conditions in hemodynamics or [32] where a least squares method is proposed for combining and enhancing the results from an existing computational fluid dynamics model with experimental data
Summary
The question of how to assimilate measured data into large scale computations of flow problems is receiving increasing attention from the computational community. This result (or more precisely its non-homogeneous version given in Corollary 2.1) will be capital in the convergence study of the numerical method presented in the sequel It is stated in [33] (with their notations, A corresponds to U and B to ∇U ): Theorem 2.1 (Conditional stability for the linearized Navier-Stokes problem) There exists R ∈ (0, 1) such that for all 0 < R1 < R2 < R3 ≤ R0 and x0 ∈ Ω satisfying R1/R3 < R2/R3 < Rand BR0(x0) ⊂ Ω, we have 1−τ τ. In a classical way for ill-posed problems [1], Corollary 2.1 gives a conditional stability result in the sense that, to be useful, this estimate has to be accompanied with an a priori bound on the solution on the global domain (due to the presence of u L in the right hand side).
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