On quasi-reversibility solutions to the Cauchy problem for the Laplace equation: regularity and error estimates

Bourgeois, Laurent; Chesnel, Lucas

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 54 (2020) no. 2, pp. 493-529.

Suite au passage du modèle économique de la revue en S20, le texte intégral des articles des années 2019 et 2020 est accessible uniquement sur le site de l'éditeur et est réservé aux abonnés.

We are interested in the classical ill-posed Cauchy problem for the Laplace equation. One method to approximate the solution associated with compatible data consists in considering a family of regularized well-posed problems depending on a small parameter ε > 0. In this context, in order to prove convergence of finite elements methods, it is necessary to get regularity results of the solutions to these regularized problems which hold uniformly in ε. In the present work, we obtain these results in smooth domains and in 2D polygonal geometries. In presence of corners, due to the particular structure of the regularized problems, classical techniques à la Grisvard do not work and instead, we apply the Kondratiev approach. We describe the procedure in detail to keep track of the dependence in ε in all the estimates. The main originality of this study lies in the fact that the limit problem is ill-posed in any framework.

G. Alessandrini, L. Rondi, E. Rosset and S. Vessella, The stability for the Cauchy problem for elliptic equations. Inverse Prob. 25 (2009) 123004. | DOI | MR | Zbl

M. Azaïez, F.B. Belgacem and H. El Fekih, On Cauchy’s problem: II. Completion, regularization and approximation. Inverse Prob. 22 (2006) 1307. | DOI | MR | Zbl

F.B. Belgacem, Why is the Cauchy problem severely ill-posed?. Inverse Prob. 23 (2007) 823–836. | DOI | MR | Zbl

L. Bourgeois, A mixed formulation of quasi-reversibility to solve the Cauchy problem for Laplace’s equation. Inverse Prob. 21 (2005) 1087–1104. | DOI | MR | Zbl

L. Bourgeois, About stability and regularization of ill-posed elliptic Cauchy problems: the case of $C^{1,1}$ domains. M2AN 44 (2010) 715–735. | DOI | Numdam | MR | Zbl

L. Bourgeois and J. Dardé, About stability and regularization of ill-posed elliptic Cauchy problems: the case of Lipschitz domains. Appl. Anal. 89 (2010) 1745–1768. | DOI | MR | Zbl

L. Bourgeois and A. Recoquillay, A mixed formulation of the Tikhonov regularization and its application to inverse PDE problems. ESAIM: M2AN 52 (2018) 123–145. | DOI | Numdam | MR

H. Brezis, Analyse fonctionnelle: théorie et applications, Editions Dunod (1999). | Zbl

E. Burman, Stabilized finite element methods for nonsymmetric, noncoercive, and ill-posed problems. Part I: Elliptic equations, SIAM J. Sci. Comput. 35 (2013) A2752–A2780. | DOI | MR | Zbl

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

E. Burman, A stabilized nonconforming finite element method for the elliptic Cauchy problem, Math. Comput. 86 (2017) 75–96. | DOI | MR

E. Burman, P. Hansbo and M.G. Larson, Solving ill-posed control problems by stabilized finite element methods: an alternative to Tikhonov regularization, Inverse Prob. 34 (2018) 035004. | DOI | MR

E. Burman, M.G. Larson and L. Oksanen, Primal-dual mixed finite element methods for the elliptic Cauchy problem, SIAM J. Numer. Anal. 56 (2018) 3480–3509. | DOI | MR

L. Chesnel and P. Ciarlet Jr, T-coercivity and continuous Galerkin methods: application to transmission problems with sign changing coefficients, Numer. Math. 124 (2013) 1–29. | DOI | MR | Zbl

L. Chesnel, X. Claeys and S.A. Nazarov, A curious instability phenomenon for a rounded corner in presence of a negative material, Asymp. Anal. 88 (2014) 43–74. | MR | Zbl

L. Chesnel, X. Claeys and S.A. Nazarov, Oscillating behaviour of the spectrum for a plasmonic problem in a domain with a rounded corner, M2AN 52 (2018) 1285–1313. | DOI | Numdam | MR

P.G. Ciarlet, The finite element method for elliptic problems. In: Vol. 4 of Studies in Mathematics and its Applications. North-Holland Publishing Co., Amsterdam-New York-Oxford (1978). | MR | Zbl

M. Costabel and M. Dauge, A singularly perturbed mixed boundary value problem, Comm. Partial Differ. Equ. 21 (1996) 1919–1949. | DOI | MR | Zbl

J. Dardé, Iterated quasi-reversibility method applied to elliptic and parabolic data completion problems, Inverse Probl. Imaging 10 (2016) 379–407. | DOI | MR

J. Dardé, A. Hannukainen and N. Hyvönen, An $H_{\mathrm{div}}$-based mixed quasi-reversibility method for solving elliptic Cauchy problems, SIAM J. Numer. Anal. 51 (2013) 2123–2148. | DOI | MR | Zbl

P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, London, 1985. | MR | Zbl

P. Grisvard, Singularities in boundary value problems. In: Vol. 22 of Recherches en Mathématiques Appliquées [Research in Applied Mathematics]. Masson, Paris; Springer-Verlag, Berlin (1992). | MR | Zbl

J. Hadamard, Sur les problèmes aux dérivées partielles et leur signification physique, Princeton Univ. Bull. (1902) 49–52.

A.M. Il’In, Matching of asymptotic expansions of solutions of boundary value problems. In: Vol. 102 of Translations of Mathematical Monographs. AMS, Providence, RI (1992). | DOI | Zbl

T. Johansson, An iterative procedure for solving a Cauchy problem for second order elliptic equations, Math. Nachr. 272 (2004) 46–54. | DOI | MR | Zbl

A. Kirsch, The Robin problem for the Helmholtz equation as a singular perturbation problem, Numer. Func. Anal. Opt. 8 (1985) 1–20. | DOI | MR | Zbl

M.V. Klibanov and F. Santosa, A computational quasi-reversibility method for Cauchy problems for Laplace’s equation, SIAM J. Appl. Math. 51 (1991) 1653–1675. | DOI | MR | Zbl

V.A. Kondratiev, Boundary-value problems for elliptic equations in domains with conical or angular points, Trans. Moscow Math. Soc. 16 (1967) 227–313. | MR | Zbl

V.A. Kozlov, V.G. Maz’Ya and J. Rossmann, Elliptic boundary value problems in domains with point singularities. In: Vol. 52 of Mathematical Surveys and Monographs. AMS, Providence, 1997. | MR | Zbl

V.A. Kozlov, V.G. Maz’Ya and J. Rossmann, Spectral problems associated with corner singularities of solutions to elliptic equations. In: Vol. 85 of Mathematical Surveys and Monographs, AMS, Providence, 2001. | MR | Zbl

R. Lattès and J.-L. Lions, Méthode de quasi-réversibilité et applications. Travaux et Recherches Mathématiques, No. 15. Dunod, Paris (1967). | MR | Zbl

V.G. Maz’Ya and B.A. Plamenevskiĭ, On the coefficients in the asymptotics of solutions of elliptic boundary value problems with conical points. Math. Nachr. 76 (1977) 29–60. Engl. transl. Amer. Math. Soc. Transl. 123 (1984) 57–89. | MR | Zbl

V.G. Maz’Ya, S.A. Nazarov and B.A. Plamenevskiĭ, In: Vol. 1 of Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains. Birkhäuser, Basel (2000). Translated from the original German 1991 edition.

S.A. Nazarov and B.A. Plamenevskiĭ, Elliptic problems in domains with piecewise smooth boundaries. In: Vol. 13 of Expositions in Mathematics. De Gruyter, Berlin, Germany, 1994. | MR | Zbl

S.A. Nazarov and N. Popoff, Self-adjoint and skew-symmetric extensions of the Laplacian with singular Robin boundary condition. C. R. Acad. Sci. Paris Ser. I 356 (2018) 927–932. | DOI | MR

S.A. Nazarov, N. Popoff and J. Taskinen, Plummeting and blinking eigenvalues of the Robin Laplacian in a cuspidal domain. Preprint (2018). | arXiv | MR

S. Nicaise, Regularity of the solutions of elliptic systems in polyhedral domains. Bull. Belg. Math. Soc. Simon Stevin 4 (1997) 411–429. | DOI | MR | Zbl

L.E. Payne, Bounds in the Cauchy problem for the Laplace equation. Arch. Ration. Mech. Anal. 5 (1960) 35–45. | DOI | MR | Zbl

L.E. Payne, On a priori bounds in the Cauchy problem for elliptic equations. SIAM J. Math. Anal. 1 (1970) 82–89. | DOI | MR | Zbl

K.-D. Phung, Remarques sur l’observabilité pour l’équation de Laplace. ESAIM Control Optim. Calc. Var. 9 (2003) 621–635. | DOI | Numdam | MR | Zbl