## 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

?

## 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. (2019). https://doi.org/10.1007/s00211-019-01087-x

## About this paper

##### Print ISSN

##### Online ISSN

## References

## References

[1] R. Becker and B. Vexler.* Optimal control of the convection-diffusion equation using stabilized finite element methods.* Numer. Math., 106(3):349–367, 2007.

[2] S. C. Brenner. *Poincare-Friedrichs inequalities for piecewise H1 functions*. SIAM J. Numer. Anal., 41(1):306–324, 2003.

[3] E. Burman. *A unified analysis for conforming and nonconforming stabilized finite element methods using interior penalty*. SIAM J. Numer. Anal., 43(5):2012–2033, 2005.

[4] E. Burman. *Stabilized finite element methods for nonsymmetric, noncoercive, and ill-posed problems. Part I: elliptic equations.* SIAM J. Sci. Comput., 35(6):A2752 A2780, 2013.

[5] E. Burman. *Error estimates for stabilized finite element methods applied to ill-posed problems*. C. R. Math. Acad. Sci. Paris, 352(7-8):655–659, 2014.

[6] E. Burman, P. Hansbo, and M. G. Larson. *Solving ill-posed control problems by stabilized finite **element methods: an alternative to Tikhonov regularization*. Inverse Problems, 34:035004, 2018.

[7] E. Burman, M. Nechita, and L. Oksanen. A stabilized finite element method for inverse problems subject to the convection–diffusion equation. II: convection dominated regime. in preparation, 2018.

[8] E. Burman, M. Nechita, and L. Oksanen. *Unique continuation for the Helmholtz equation using stabilized finite element methods*. J. Math. Pures Appl., 2018.

[9] E. Burman and L. Oksanen. *Data assimilation for the heat equation using stabilized finite element methods.* Numer. Math., 139(3):505–528, 2018.

[10] L. Dede’ and A. Quarteroni. *Optimal control and numerical adaptivity for advection-diffusion equations*. M2AN Math. Model. Numer. Anal., 39(5):1019–1040, 2005.

[11] D. Dos Santos Ferreira, C. E. Kenig, M. Salo, and G. Uhlmann. *Limiting Carleman weights and anisotropic inverse problems*. Invent. Math., 178(1):119–171, 2009.

[12] A. Ern and J.-L. Guermond. *Theory and practice of finite elements*, volume 159 of Applied Mathematical Sciences. Springer-Verlag, New York, 2004.

[13] A. Ern and J.-L. Guermond. *Evaluation of the condition number in linear systems arising in finite element approximations*. M2AN Math. Model. Numer. Anal., 40(1):29 48, 2006.

[14] F. Hecht. *New development in FreeFem++. J*. Numer. Math., 20(3-4):251–265, 2012.

[15] M. Hinze, N. Yan, and Z. Zhou. Variational discretization for optimal control governed by convection dominated diffusion equations. J. Comput. Math., 27(2-3):237–253, 2009.

[16] J. Le Rousseau and G. Lebeau. *On Carleman estimates for elliptic and parabolic operators*. Applications to unique continuation and control of parabolic equations. ESAIM Control Optim. Calc. Var., 18(3):712–747, 2012.

[17] E. Malinnikova and S. Vessella. *Quantitative uniqueness for elliptic equations with singular lower order terms*. Math. Ann., 353(4):1157–1181, 2012.

[18] P. Monk and E. Suli. *The adaptive computation of far-field patterns by a posteriori error estimation of linear functionals*. SIAM J. Numer. Anal., 36(1):251–274, 1999.

[19] N. Yan and Z. Zhou. *A priori and a posteriori error analysis of edge stabilization Galerkin method for **the optimal control problem governed by convection-dominated diffusion equation*. J. Comput. Appl. Math., 223(1):198–217, 2009.

[20] M. Zworski. *Semiclassical analysis, volume 138 of Graduate Studies in Mathematics*. American Mathematical Society, Providence, RI, 2012.[32] M.D. Zeiler, ADADELTA : *An adaptive learning rate method*, 2012,http://arxiv.org/abs/1212.5701.