## 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.

## 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, convection-dominated, unique continuation, ill-posed problem

## Paper coordinates

E. Burman, M. Nechita, L. Oksanen, *A stabilized finite element method for inverse problems subject to the convection-diffusion equation. II: convection-dominated regime, *Numer. Math., 150:769-801, 2022, DOI: https://doi.org/10.1007/s00211-022-01268-1

Open Access: https://doi.org/10.1007/s00211-022-01268-1

## About this paper

##### Print ISSN

0029-599X

##### Online ISSN

0945-3245

[1] M. S. Alnæs, J. Blechta, J. Hake, A. Johansson, B. Kehlet, A. Logg, C. Richardson, J. Ring, M. E. Rognes, and G. N. Wells. The FEniCS Project Version 1.5. Archive of Numerical Software, 3(100), 2015.

[2] S. Bertoluzza. The discrete commutator property of approximation spaces. C. R. Acad. Sci.Paris Ser. I Math., 329(12):1097–1102, 1999.

[3] E. Burman, J. Guzman, and D. Leykekhman. Weighted error estimates of the continuous interior penalty method for singularly perturbed problems. IMA J. Numer. Anal., 29(2):284– 314, 2009.

[4] 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.

[5] E. Burman, M. Nechita, and L. Oksanen. A stabilized finite element method for inverse problems subject to the convection–diffusion equation. I: diffusion-dominated regime. Numer. Math., 144(451–477), 2020.

[6] 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.

[7] 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.

[8] E. Burman. Stabilized finite element methods for nonsymmetric, noncoercive, and ill-posed problems. Part II: Hyperbolic equations. SIAM J. Sci. Comput., 36(4):A1911–A1936, 2014.

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

[10] L. C. Evans. Partial differential equations, volume 19 of Graduate Studies in Mathematics. American Mathematical Society, 2010.

[11] V. John, P. Knobloch, and J. Novo. Finite elements for scalar convection-dominated equations and incompressible flow problems: a never ending story? Comput. Vis. Sci., 19(5):47– 63, Dec 2018.

[12] C. Johnson, U. Navert, and J. Pitkaranta. Finite element methods for linear hyperbolic problems. Comput. Methods Appl. Mech. Engrg., 45(1-3):285–312, 1984.

[13] 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.

[14] H.-G. Roos and M. Stynes. Some open questions in the numerical analysis of singularly perturbed differential equations. Comput. Methods Appl. Math., 15(4):531–550, 2015.

[15] E. M Stein. Singular integrals and differentiability properties of functions, volume 30 of Princeton Mathematical Series. Princeton University Press, 1970.