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

TL;DR

This paper extends a stabilized finite element method to the convection-dominated regime for inverse problems involving stationary convection-diffusion equations, achieving quasi-optimal convergence and analyzing data perturbations.

Contribution

The paper adapts and analyzes a stabilized finite element method specifically for the convection-dominated regime in inverse problems, including error analysis and numerical validation.

Findings

Quasi-optimal convergence along characteristics

Effective handling of data perturbations

Numerical experiments confirming theoretical results

Abstract

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.