We consider the numerical approximation of the ill-posed data assimilation problem for stationary convection-diffusion equations and extend our previous analysis in [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 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.

中文翻译：

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受对流-扩散方程影响的反问题的稳定有限元方法。I: 扩散主导体制

我们考虑了平稳对流扩散方程不适定数据同化问题的数值近似，并扩展了我们之前在 [Numer. 数学。144, 451--477, 2020] 到对流主导的制度。稍微调整为主导扩散提出的稳定有限元方法，我们利用局部误差分析，通过数据集沿着对流场的特征获得准最优收敛。将乘以离散解的权重函数取为 Lipschitz，并证明了相应的超近似结果（离散换向器性质）。数据扰动的影响包含在分析中，我们用一些数值实验结束了这篇论文。