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Unique continuation for an elliptic interface problem using unfitted isoparametric finite elements

Burman, E; Preuss, J; (2025) Unique continuation for an elliptic interface problem using unfitted isoparametric finite elements. Smai Journal of Computational Mathematics , 11 pp. 165-202. 10.5802/smai-jcm.122. Green open access

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Abstract

We study unique continuation over an interface using a stabilized unfitted finite element method tailored to the conditional stability of the problem. The interface is approximated using an isoparametric transformation of the background mesh and the corresponding geometrical error is included in our error analysis. To counter possible destabilizing effects caused by non-conformity of the discretization and cope with the interface conditions, we introduce adapted regularization terms. This allows to derive error estimates based on conditional stability. The necessity and effectiveness of the regularization is illustrated in numerical experiments. We also explore numerically the effect of the heterogeneity in the coefficients on the ability to reconstruct the solution outside the data domain. For Helmholtz equations we find that a jump in the flux impacts the stability of the problem significantly less than the size of the wavenumber.

Type: Article
Title: Unique continuation for an elliptic interface problem using unfitted isoparametric finite elements
Open access status: An open access version is available from UCL Discovery
DOI: 10.5802/smai-jcm.122
Publisher version: https://doi.org/10.5802/smai-jcm.122
Language: English
Additional information: © The authors, 2025. https://creativecommons.org/licenses/by/4.0/
Keywords: unfitted finite element method, unique continuation, interface problems, isoparametric finite element method, geometry errors, conditional Hölder stability
UCL classification: UCL
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics
URI: https://discovery.ucl.ac.uk/id/eprint/10211253
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