In this work we consider the computational approximation of a unique continuation problem for the Helmholtz equation using a stabilized finite element method. First conditional stability estimates are derived for which, under a convexity assumption on the geometry, the constants grow at most linearly in the wave number. Then these estimates are used to obtain error bounds for the finite element method that are explicit with respect to the wave number. Some numerical illustrations are given.


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

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

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


Helmholtz equation; unique continuation; Finite Element method; wave number explicit; conditional Hölder stability.

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E. Burman, M. Nechita, L. Oksanen, Unique continuation for the Helmholtz equation using stabilized finite element methods, J. Math. Pures Appl., 129 (2019), pp. 1-22.
DOI: 10.1016/j.matpur.2018.10.003


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Journal de Mathématiques Pures et Appliquées

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[1] G. Alessandrini, L. Rondi, E. Rosset, S. Vessella, The stability for the Cauchy problem for elliptic equations, Inverse Problems, 25 (2009), Article 123004

[2] I.M. Babuška, S.A. Sauter, Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers SIAM J. Numer. Anal., 34 (6) (1997), pp. 2392-2423

[3] D. Baskin, E.A. Spence, J. Wunsch, Sharp high-frequency estimates for the Helmholtz equation and applications to boundary integral equations, SIAM J. Math. Anal., 48 (1) (2016), pp. 229-267

[4] L. Bourgeois, J. Dardé, About stability and regularization of ill-posed elliptic Cauchy problems: the case of Lipschitz domains, Appl. Anal., 89 (11) (2010), pp. 1745-1768

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

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

[7] E. Burman, Stabilised finite element methods for ill-posed problems with conditional stability, Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations, Lect. Notes Comput. Sci. Eng., vol. 114, Springer (2016), pp. 93-127

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

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

[10] E. Burman, L. Oksanen, Data assimilation for the heat equation using stabilized finite element methods, Numer. Math., 139 (3) (2018), pp. 505-528

[11] E. Burman, H. Wu, L. Zhu, Linear continuous interior penalty finite element method for Helmholtz equation with high wave number: one-dimensional analysis, Numer. Methods Partial Differ. Equ., 32 (5) (2016), pp. 1378-1410

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

[13] A. Ern, J.-L. Guermond, Theory and Practice of Finite Elements, Appl. Math. Sci., vol. 159, Springer-Verlag, New York (2004)

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

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