Comptes Rendus
no. 4
Numerical analysis
The elliptic Cauchy problem revisited: Control of boundary data in natural norms
[Le problème de Cauchy pour les équations elliptiques revisité : contrôle frontière en normes naturelles]

Présenté par : Olivier Pironneau

Erik Burman1
1 Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, United Kingdom
Comptes Rendus. Mathématique, Volume 355 (2017) no. 4, pp. 479-484.

Dans cette note, nous montrons des estimations d'erreur pour l'approximation d'éléments finis des données sur le bord d'un problème de Cauchy elliptique. Ces résultats complètent l'analyse d'erreur de la méthode d'éléments finis proposée dans E. Burman, Error estimates for stabilized finite element methods applied to ill-posed problems, C. R. Acad. Sci. Paris, Ser. I 352 (7–8) (2014) 655–659.

In this note, we prove error estimates in natural norms on the approximation of the boundary data in the elliptic Cauchy problem, for the finite element method first analysed in E. Burman, Error estimates for stabilised finite element methods applied to ill-posed problems, C. R. Acad. Sci. Paris, Ser. I 352 (7–8) (2014) 655–659.

Reçu le :
Accepté le :
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DOI : 10.1016/j.crma.2017.02.014
Erik Burman 1

1 Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, United Kingdom 
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Erik Burman. The elliptic Cauchy problem revisited: Control of boundary data in natural norms. Comptes Rendus. Mathématique, Volume 355 (2017) no. 4, pp. 479-484. doi : 10.1016/j.crma.2017.02.014. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.02.014/

[1] G. Alessandrini; L. Rondi; E. Rosset; S. Vessella The stability for the Cauchy problem for elliptic equations, Inverse Probl., Volume 25 (2009) no. 12

[2] S.C. Brenner Poincaré–Friedrichs inequalities for piecewise H1 functions, SIAM J. Numer. Anal., Volume 41 (2003) no. 1, pp. 306-324

[3] E. Burman A penalty free nonsymmetric Nitsche-type method for the weak imposition of boundary conditions, SIAM J. Numer. Anal., Volume 50 (2012), pp. 1959-1981

[4] E. Burman Stabilized finite element methods for nonsymmetric, noncoercive, and ill-posed problems. Part I: Elliptic equations, SIAM J. Sci. Comput., Volume 35 (2013) no. 6, p. A2752-A2780

[5] E. Burman Error estimates for stabilized finite element methods applied to ill-posed problems, C. R. Acad. Sci. Paris, Ser. I, Volume 352 (2014) no. 7–8, pp. 655-659

[6] 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, LMS Proceedings, Springer, 2014

[7] E. Burman A stabilized nonconforming finite element method for the elliptic Cauchy problem, Math. Comput., Volume 86 (2017) no. 303, pp. 75-96

[8] O. Steinbach Numerical Approximation Methods for Elliptic Boundary Value Problems, Finite and Boundary Elements, Springer, New York, 2008

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