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The numerical approximation of an inverse problem subject to theconvection--diffusion equation when diffusion dominates is studied. We deriveCarleman estimates that are on a form suitable for use in numerical analysisand with explicit dependence on the P\'eclet number. A stabilized finiteelement method is then proposed and analysed. An upper bound on the conditionnumber is first derived. Combining the stability estimates on the continuousproblem with the numerical stability of the method, we then obtain errorestimates in local $H^1$- or $L^2$-norms that are optimal with respect to theapproximation order, the problem's stability and perturbations in data. Theconvergence order is the same for both norms, but the $H^1$-estimate requiresan additional divergence assumption for the convective field. The theory isillustrated in some computational examples.

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