## Abstract

We consider the numerical approximation of the linear ill-posed problem of unique continuation for the Helmholtz equation. We first review the conditional stability of this problem and then discuss high-order conforming finite element methods, with regularization added on the discrete level using gradient jump penalty and Galerkin least squares. The method is shown to converge in terms of the stability of the problem, quantified by the Hölder exponent, and the polynomial degree of approximation. The analysis also takes into account possibly noisy data. Numerical examples illustrating the theory are presented for a Helmholtz version of the classical Hadamard example for two geometric configurations with different stability properties for the continuation problem, including data perturbations of different amplitudes.

## Authors

Mihai Nechita

“Tiberiu Popoviciu” Institute of Numerical Analysis, Romanian Academy

## Keywords

## About this paper

##### Cite this paper as:

##### Journal

##### Publisher Name

##### DOI

Not available yet.

##### Print ISSN

Not available yet.

##### Online ISSN

Not available yet.

##### Google Scholar Profile

[1] E. Burman, G. Delay, and A. Ern., *A hybridized high-order method for unique continuation subject to the helmholtz equation*, SIAM J. Numer. Anal., 59(5):2368–2392,2021.[BHL18] E. Burman, P. Hansbo, and M. G. Larson, *Solving ill-posed control problems by **stabilized finite element methods: an alternative to Tikhonov regularization*, Inverse Problems, 34:035004, 2018.

[2] S. M. Berge and E. Malinnikova, *On the three ball theorem for solutions of the **Helmholtz equation*, Complex Anal. Synerg., 7(2):1–10, 2021.

[3] E. Burman, M. Nechita, and L. Oksanen, *Unique continuation for the Helmholtz **equation using stabilized finite element methods*, J. Math. Pures Appl., 129:1–22, 2019.

[4] I. M. Babuska and S. A. Sauter, *Is the pollution effect of the FEM avoidable for **the Helmholtz equation considering high wave numbers?* SIAM J. Numer. Anal., 34(6):2392–2423, 1997.

[5] E. Burman, *Stabilized finite element methods for nonsymmetric, noncoercive, and **ill-posed problems. Part I: Elliptic equations*, SIAM J, Sci. Comput., 35(6):A2752–

A2780, 2013.

[6] E. Burman, *Stabilised finite element methods for ill-posed problems with conditional **stability. In Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations,* volume 114 of Lect. Notes Comput. Sci. Eng., pages 93–127. Springer, 2016.

[7] A. Ern and J.-L. Guermond, *Finite elements I: Approximation and interpolation, **volume 72 of Texts in Applied Mathematics*, Springer Nature, 2021.

[8] T. Hrycak and V. Isakov, *Increased stability in the continuation of solutions to the **Helmholtz equation*, Inverse Problems, 20(3):697–712, 2004.

[9] V. Isakov, *Inverse problems for partial differential equations, volume 127 of Applied **Mathematical Sciences*, Springer, 3rd edition, 2017.

[10] J. M. Melenk and S. A. Sauter, *Wavenumber explicit convergence analysis for Galerkin **discretizations of the Helmholtz equation*, SIAM J. Numer. Anal., 49(3):1210–1243, 2011.

[11] M. Nechita, *Unique continuation problems and stabilised finite element methods*, PhD thesis, University College London, 2020.