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Unique continuation for an elliptic interface problem using unfitted isoparametric finite elements
Erik Burman1; Janosch Preuss1
1 Department of Mathematics, University College London, United Kingdom
The SMAI Journal of computational mathematics, Volume 11 (2025), pp. 165-202.

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.

Published online:
DOI: 10.5802/smai-jcm.122
Classification: 35J15, 65N12, 65N20, 65N30, 86-08
Keywords: unfitted finite element method, unique continuation, interface problems, isoparametric finite element method, geometry errors, conditional Hölder stability

Erik Burman 1; Janosch Preuss 1

1 Department of Mathematics, University College London, United Kingdom
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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     title = {Unique continuation for an elliptic interface problem using unfitted isoparametric finite elements},
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Erik Burman; Janosch Preuss. Unique continuation for an elliptic interface problem using unfitted isoparametric finite elements. The SMAI Journal of computational mathematics, Volume 11 (2025), pp. 165-202. doi : 10.5802/smai-jcm.122. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.122/

[1] Giovanni Alessandrini; Luca Rondi; Edi Rosset; Sergio Vessella The stability for the Cauchy problem for elliptic equations, Inverse Probl., Volume 25 (2009) no. 12, 123004, 47 pages | DOI | MR | Zbl

[2] Nick Alger; Umberto Villa; Tan Bui-Thanh; Omar Ghattas A data scalable augmented Lagrangian KKT preconditioner for large-scale inverse problems, SIAM J. Sci. Comput., Volume 39 (2017) no. 5, p. A2365-A2393 | DOI | MR | Zbl

[3] Christine Bernardi Optimal Finite-Element Interpolation on Curved Domains, SIAM J. Numer. Anal., Volume 26 (1989) no. 5, pp. 1212-1240 | DOI | MR | Zbl

[4] Thomas Boiveau; Erik Burman; Susanne Claus; Mats Larson Fictitious domain method with boundary value correction using penalty-free Nitsche method, J. Numer. Math., Volume 26 (2018) no. 2, pp. 77-95 | DOI | MR | Zbl

[5] Muriel Boulakia; Erik Burman; Miguel A. Fernández; Colette Voisembert Data assimilation finite element method for the linearized Navier–Stokes equations in the low Reynolds regime, Inverse Probl., Volume 36 (2020) no. 8, 085003, 22 pages | DOI | MR | Zbl

[6] Erik 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 | DOI | MR | Zbl

[7] Erik Burman; Susanne Claus; Peter Hansbo; Mats Larson; André Massing CutFEM: discretizing geometry and partial differential equations, Int. J. Numer. Methods Eng., Volume 104 (2015) no. 7, pp. 472-501 | DOI | MR | Zbl

[8] Erik Burman; Guillaume Delay; Alexandre Ern A Hybridized High-Order Method for Unique Continuation Subject to the Helmholtz Equation, SIAM J. Numer. Anal., Volume 59 (2021) no. 5, pp. 2368-2392 | DOI | MR | Zbl

[9] Erik Burman; Alexandre Ern An Unfitted Hybrid High-Order Method for Elliptic Interface Problems, SIAM J. Numer. Anal., Volume 56 (2018) no. 3, pp. 1525-1546 | DOI | MR | Zbl

[10] Erik Burman; Ali Feizmohammadi; Arnaud Münch; Lauri Oksanen Space time stabilized finite element methods for a unique continuation problem subject to the wave equation, ESAIM, Math. Model. Numer. Anal., Volume 55 (2021), p. S969-S991 | DOI | MR | Zbl

[11] Erik Burman; Ali Feizmohammadi; Arnaud Münch; Lauri Oksanen Spacetime finite element methods for control problems subject to the wave equation, ESAIM, Control Optim. Calc. Var., Volume 29 (2023), 41, 40 pages | DOI | MR | Zbl

[12] Erik Burman; Johnny Guzmán; Manuel A Sánchez; Marcus Sarkis Robust flux error estimation of an unfitted Nitsche method for high-contrast interface problems, IMA J. Numer. Anal., Volume 38 (2017) no. 2, pp. 646-668 | DOI | MR | Zbl

[13] Erik Burman; Peter Hansbo; Mats Larson A cut finite element method with boundary value correction, Math. Comput., Volume 87 (2018) no. 310, pp. 633-657 | DOI | MR | Zbl

[14] Erik Burman; Peter Hansbo; Mats Larson Solving ill-posed control problems by stabilized finite element methods: an alternative to Tikhonov regularization, Inverse Probl., Volume 34 (2018) no. 3, 035004 | DOI | MR | Zbl

[15] Erik Burman; Mihai Nechita; Lauri Oksanen Unique continuation for the Helmholtz equation using stabilized finite element methods, J. Math. Pures Appl., Volume 129 (2019), pp. 1-22 | DOI | MR | Zbl

[16] Erik Burman; Mihai Nechita; Lauri Oksanen A stabilized finite element method for inverse problems subject to the convection–diffusion equation. I: diffusion-dominated regime, Numer. Math., Volume 144 (2020) no. 3, pp. 451-477 | DOI | MR | Zbl

[17] Erik Burman; Mihai Nechita; Lauri Oksanen A stabilized finite element method for inverse problems subject to the convection–diffusion equation. II: convection-dominated regime, Numer. Math., Volume 150 (2022) no. 3, pp. 769-801 | DOI | MR | Zbl

[18] Erik Burman; Janosch Preuss Reproduction material: Unique continuation for an elliptic interface problem using unfitted isoparametric finite elements (v2) | DOI

[19] Erik Burman; Janosch Preuss Unique continuation for the Lamé system using stabilized finite element methods, GEM. Int. J. Geomath., Volume 14 (2023) no. 1, 9, 36 pages | DOI | MR | Zbl

[20] Erik Burman; Paolo Zunino A Domain Decomposition Method Based on Weighted Interior Penalties for Advection‐Diffusion‐Reaction Problems, SIAM J. Numer. Anal., Volume 44 (2006) no. 4, pp. 1612-1638 | DOI | MR | Zbl

[21] Cătălin I. Cârstea; Jenn-Nan Wang Propagation of smallness for an elliptic PDE with piecewise Lipschitz coefficients, J. Differ. Equations, Volume 268 (2020) no. 12, pp. 7609-7628 | DOI | MR | Zbl

[22] Zhiming Chen; Ke Li; Xueshuang Xiang An adaptive high-order unfitted finite element method for elliptic interface problems, Numer. Math., Volume 149 (2021) no. 3, pp. 507-548 | DOI | MR | Zbl

[23] Wolfgang Dahmen; Harald Monsuur; Rob Stevenson Least squares solvers for ill-posed PDEs that are conditionally stable, ESAIM, Math. Model. Numer. Anal., Volume 57 (2023) no. 4, pp. 2227-2255 | DOI | MR | Zbl

[24] Maksymilian Dryja On Discontinuous Galerkin Methods for Elliptic Problems with Discontinuous Coefficients, Comput. Methods Appl. Math., Volume 3 (2003) no. 1, pp. 76-85 | DOI | MR | Zbl

[25] Lars Eldén; Valeria Simoncini A numerical solution of a Cauchy problem for an elliptic equation by Krylov subspaces, Inverse Probl., Volume 25 (2009) no. 6, 065002, 22 pages | MR | Zbl

[26] Alexandre Ern; Annette F. Stephansen; Paolo Zunino A discontinuous Galerkin method with weighted averages for advection–diffusion equations with locally small and anisotropic diffusivity, IMA J. Numer. Anal., Volume 29 (2009) no. 2, pp. 235-256 | DOI | MR | Zbl

[27] Gerald B Folland Real analysis: modern techniques and their applications, Pure and Applied Mathematics, John Wiley & Sons, 1999 | MR

[28] Roland Glowinski; Chin-Hsien Li; Jacques-Louis Lions A numerical approach to the exact boundary controllability of the wave equation (I) Dirichlet controls: Description of the numerical methods, Jpn. J. Appl. Math., Volume 7 (1990), pp. 1-76 | DOI | Zbl

[29] Jörg Grande; Christoph Lehrenfeld; Arnold Reusken Analysis of a High-Order Trace Finite Element Method for PDEs on Level Set Surfaces, SIAM J. Numer. Anal., Volume 56 (2018) no. 1, pp. 228-255 | DOI | MR | Zbl

[30] Anita Hansbo; Peter Hansbo An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems, Comput. Methods Appl. Mech. Eng., Volume 191 (2002) no. 47-48, pp. 5537-5552 | DOI | MR | Zbl

[31] Christoph Lehrenfeld High order unfitted finite element methods on level set domains using isoparametric mappings, Comput. Methods Appl. Mech. Eng., Volume 300 (2016), pp. 716-733 | DOI | MR | Zbl

[32] Christoph Lehrenfeld A Higher Order Isoparametric Fictitious Domain Method for Level Set Domains, Geometrically Unfitted Finite Element Methods and Applications (Stéphane P. A. Bordas; Erik Burman; Mats Larson; Maxim A. Olshanskii, eds.), Springer, 2017, pp. 65-92 | DOI | Zbl

[33] Christoph Lehrenfeld; Fabian Heimann; Janosch Preuß; Henry von Wahl ngsxfem: Add-on to NGSolve for geometrically unfitted finite element discretizations, J. Open Source Softw., Volume 6 (2021) no. 64, 3237 | DOI

[34] Christoph Lehrenfeld; Arnold Reusken Analysis of a high order unfitted finite element method for an elliptic interface problem, IMA J. Numer. Anal., Volume 38 (2018) no. 3, pp. 1351-1387 | DOI | MR | Zbl

[35] Christoph Lehrenfeld; Arnold Reusken L 2 -estimates for a high order unfitted finite element method for elliptic interface problems, J. Numer. Math., Volume 27 (2019) no. 2, pp. 85-99 | DOI | Zbl

[36] Yimin Lou High-order Unfitted Discretizations for Partial Differential Equations Coupled with Geometric Flow, Ph. D. Thesis, University of Göttingen (2023) | DOI

[37] Kent-André Mardal; Bjørn Fredrik Nielsen; Magne Nordaas Robust preconditioners for PDE-constrained optimization with limited observations, BIT, Volume 57 (2017), pp. 405-431 | DOI | MR | Zbl

[38] Ralf Massjung An Unfitted Discontinuous Galerkin Method Applied to Elliptic Interface Problems, SIAM J. Numer. Anal., Volume 50 (2012) no. 6, pp. 3134-3162 | DOI | MR | Zbl

[39] Siddhartha Mishra; Roberto Molinaro Estimates on the generalization error of physics-informed neural networks for approximating PDEs, IMA J. Numer. Anal., Volume 43 (2023) no. 1, pp. 1-43 | DOI | MR | Zbl

[40] Mihai Nechita Unique continuation problems and stabilised finite element methods, Ph. D. Thesis, University College London (2020) https://discovery.ucl.ac.uk/id/eprint/10113065

[41] Arnold Reusken Analysis of an extended pressure finite element space for two-phase incompressible flows, Comput. Vis. Sci., Volume 11 (2008) no. 4, pp. 293-305 | DOI | Zbl

[42] Luc Robbiano Théorème d’unicité adapté au contrôle des solutions des problèmes hyperboliques, Commun. Partial Differ. Equations, Volume 16 (1991) no. 4-5, pp. 789-800 | DOI | MR

[43] Joachim Schöberl NETGEN An advancing front 2D/3D-mesh generator based on abstract rules, Comput. Vis. Sci., Volume 1 (1997) no. 1, pp. 41-52 | DOI | Zbl

[44] Joachim Schöberl C++11 Implementation of Finite Elements in NGSolve (2014) https://www.asc.tuwien.ac.at/%7eschoeberl/wiki/publications/ngs-cpp11.pdf (Technical report)

[45] Elias M. Stein Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, 30, Princeton University Press, 1971 | DOI | MR

[46] Haijun Wu; Yuanming Xiao An Unfitted hp-Interface Penalty Finite Element Method for Elliptic Interface Problems, J. Comput. Math., Volume 37 (2018) no. 3, pp. 316-339 | DOI | MR | Zbl

[47] Yuanming Xiao; Jinchao Xu; Fei Wang High-order extended finite element methods for solving interface problems, Comput. Methods Appl. Mech. Eng., Volume 364 (2020), 112964, 21 pages | DOI | MR | Zbl

[48] Paolo Zunino; Laura Cattaneo; Claudia Maria Colciago An unfitted interface penalty method for the numerical approximation of contrast problems, Appl. Numer. Math., Volume 61 (2011) no. 10, pp. 1059-1076 | DOI | MR | Zbl

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