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

## Abstract

The numerical approximation of an inverse problem subject to the convection–diffusion equation when diffusion dominates is studied. We derive Carleman estimates that are on a form suitable for use in numerical analysis and with explicit dependence on the Peclet number. A stabilized finite element method is then proposed and analysed. An upper bound on the condition number is first derived. Combining the stability estimates on the continuous problem with the numerical stability of the method, we then obtain error estimates in local H1- or L2-norms that are optimal with respect to the approximation order, the problem’s stability and perturbations in data. The convergence order is the same for both norms, but the H1-estimate requires an additional divergence assumption for theconvective field. The theory is illustrated in some computational examples

## Authors

Erik Burman
Department of Mathematics, University College London, Gower Street, London UK, WC1E 6BT

Mihai Nechita
Department of Mathematics, University College London, Gower Street, London UK, WC1E 6BT

Lauri Oksanen
Department of Mathematics, University College London, Gower Street, London UK, WC1E 6BT

## Keywords

convection-diffusion equation; FEM (Finite Element Method);

## Paper coordinates

E. Burman, M. Nechita, L. Oksanen, A stabilized finite element method for inverse problems subject to the convection-diffusion equation. I: diffusion-dominated regime, Numer. Math. 144 (2020), pp. 451-477.
https://doi.org/10.1007/s00211-019-01087-x

## PDF

##### Journal

Numerische Mathematik

Springer

0029-599X

0945-3245