Volume 57, Number 6, November-December 2023
|3499 - 3536
|20 December 2023
Stable approximation of Helmholtz solutions in the disk by evanescent plane waves
Alpines, Inria 2 rue Simone Iff, 75012 Paris, France
2 Laboratoire Jacques-Louis Lions, Sorbonne Université & CNRS, 4 place Jussieu, 75005 Paris, France
3 Department of Computer Science, KU Leuven, Celestijnenlaan 200A, 3001 Leuven, Belgium
4 Dipartimento di Matematica, Università di Pavia, Via Ferrata 5, 27100 Pavia, Italy
Accepted: 29 September 2023
Superpositions of plane waves are known to approximate well the solutions of the Helmholtz equation. Their use in discretizations is typical of Trefftz methods for Helmholtz problems, aiming to achieve high accuracy with a small number of degrees of freedom. However, Trefftz methods lead to ill-conditioned linear systems, and it is often impossible to obtain the desired accuracy in floating-point arithmetic. In this paper we show that a judicious choice of plane waves can ensure high-accuracy solutions in a numerically stable way, in spite of having to solve such ill-conditioned systems. Numerical accuracy of plane wave methods is linked not only to the approximation space, but also to the size of the coefficients in the plane wave expansion. We show that the use of plane waves can lead to exponentially large coefficients, regardless of the orientations and the number of plane waves, and this causes numerical instability. We prove that all Helmholtz fields are continuous superposition of evanescent plane waves, i.e., plane waves with complex propagation vectors associated with exponential decay, and show that this leads to bounded representations. We provide a constructive scheme to select a set of real and complex-valued propagation vectors numerically. This results in an explicit selection of plane waves and an associated Trefftz method that achieves accuracy and stability. The theoretical analysis is provided for a two-dimensional domain with circular shape. However, the principles are general and we conclude the paper with a numerical experiment demonstrating practical applicability also for polygonal domains.
Mathematics Subject Classification: 35J05 / 41A30 / 42C15 / 44A15
Key words: Helmholtz equation / plane waves / evanescent waves / Trefftz method / stable approximation / sampling / frames / reproducing kernel Hilbert spaces / Herglotz representation
© The authors. Published by EDP Sciences, SMAI 2023
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.