Abstract
We consider the numerical approximation of the ill-posed data assimilation problem for stationary convection–diffusion equations and extend our previous analysis in Burman et al. (Numer. Math. 144:451–477, 2020) to the convection-dominated regime. Slightly adjusting the stabilized finite element method proposed for dominant diffusion, we draw upon a local error analysis to obtain quasi-optimal convergence along the characteristics of the convective field through the data set. The weight function multiplying the discrete solution is taken to be Lipschitz continuous and a corresponding super approximation result (discrete commutator property) is proven. The effect of data perturbations is included in the analysis and we conclude the paper with some numerical experiments.
Highlights
In this work, we consider a data assimilation problem for a stationary convection– diffusion equationLu := −μ u + β · ∇u = f in ⊂ Rn, (1)when convection dominates, that is 0 < μ |β|, and complement the diffusiondominated case discussed in the first part [5]
We assume that ⊂ Rn is open, bounded and connected, and there exists a solution u ∈ H 2( ) to (1)
Pe(h) 1, the goal of this second part paper is to reconsider the numerical method proposed in the first part [5] and develop an error analysis that captures and exploits the governing transport phenomenon
Summary
We consider a data assimilation problem for a stationary convection– diffusion equation. Consider an open bounded set B ⊂ that contains the data region ω such that B \ ω does not touch the boundary of. For u ∈ H 1( ), the following conditional stability estimate was proven for μ > 0 and β ∈ L∞( )n, u L2(B) ≤ Cst u. Note that the continuum estimate (2) is valid in both the diffusion-dominated and convection-dominated regimes, and that the stability constant Cst is uniformly bounded when diffusion dominates. The continuum estimate (2) was combined with a stabilized linear finite element method to obtain convergence orders for the approximate solution. Pe(h) := |β|h , μ the following error bound [5, Theorem 1] was proven for the approximation uh in the diffusive regime Pe(h) < 1, u − uh L2(B) ≤ Cst hκ ( u H2( ) + h−1 δ L2(ω)),. For the computation we used an unstructured mesh with 512 elements on a side and mesh size h ≈ 0.0025
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