A stabilized finite element method for inverse problems subject to the convection–diffusion equation. II: convection-dominated regime

• Published in: Numerische Mathematik Vol. 150; no. 3; pp. 769 - 801
• Main Authors: Burman, Erik, Nechita, Mihai, Oksanen, Lauri
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.03.2022; Springer Nature B.V
• Subjects: Approximation; Commutators; Convection-diffusion equation; Diffusion; Error analysis; Finite element analysis; Finite element method; Inverse problems; Mathematical analysis; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Numerical Analysis; Numerical and Computational Physics; Perturbation; Simulation; Theoretical; Weighting functions; 65N12; 35J15; 65N21; 65N30; 65N20
• ISSN: 0029-599X; 0945-3245
• DOI: 10.1007/s00211-022-01268-1

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.