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.