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

Laurent Bourgeois 1 Lucas Chesnel 2, 3
1 POEMS - Propagation des Ondes : Étude Mathématique et Simulation
Inria Saclay - Ile de France, UMA - Unité de Mathématiques Appliquées, CNRS - Centre National de la Recherche Scientifique : UMR7231
3 DeFI - Shape reconstruction and identification
Inria Saclay - Ile de France, CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
Abstract : 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.
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Submitted on : Thursday, November 28, 2019 - 6:26:00 PM
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Laurent Bourgeois, Lucas Chesnel. On quasi-reversibility solutions to the Cauchy problem for the Laplace equation: regularity and error estimates. ESAIM: Mathematical Modelling and Numerical Analysis, EDP Sciences, 2020, ⟨10.1051/m2an/2019073⟩. ⟨hal-02385487⟩

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