Abstract
Over the last 10 years, results from [J. M. Melenk and S. Sauter, Math. Comp., 79 (2010), pp. 1871-1914], [J. M. Melenk and S. Sauter, SIAM J. Numer. Anal., 49 (2011), pp. 1210- 1243], [S. Esterhazy and J. M. Melenk, Numerical Analysis of Multiscale Problems, Springer, New York, 2012, pp. 285-324] and [J. M. Melenk, A. Parsania, and S. Sauter, J. Sci. Comput., 57 (2013), pp. 536-581] decomposing high-frequency Helmholtz solutions into "low-" and "high-" frequency components have had a large impact in the numerical analysis of the Helmholtz equation. These results have been proved for the constant-coefficient Helmholtz equation in either the exterior of a Dirichlet obstacle or an interior domain with an impedance boundary condition. Using the Helffer- Sjöstrand functional calculus [B. Helffer and J. Sjöstrand, Schrödinger Operators, Springer, Berlin, 1989, pp. 118-197] this paper proves analogous decompositions for scattering problems fitting into the black-box scattering framework of Sjöstrand and Zworski [J. Amer. Math. Soc., 4 (1991), pp. 729-769] thus covering Helmholtz problems with variable coefficients, impenetrable obstacles, and penetrable obstacles all at once. These results allow us to prove new frequency-explicit convergence results for (i) the hp-finite-element method (hp-FEM) applied to the variable-coefficient Helmholtz equation in the exterior of an analytic Dirichlet obstacle, where the coefficients are analytic in a neighborhood of the obstacle, and (ii) the h-FEM applied to the Helmholtz penetrable-obstacle transmission problem. In particular, the result in (i) shows that the hp-FEM applied to this problem does not suffer from the pollution effect.
Original language | English (US) |
---|---|
Pages (from-to) | 3903-3958 |
Number of pages | 56 |
Journal | SIAM Journal on Mathematical Analysis |
Volume | 55 |
Issue number | 4 |
DOIs | |
State | Published - 2023 |
Funding
*Received by the editors April 1, 2021; accepted for publication (in revised form) February 24, 2023; published electronically August 22, 2023. https://doi.org/10.1137/21M1409160 Funding: The work of the first author was supported by EPSRC grant EP/V001760/1. The work of the second and third authors was supported by EPSRC grant EP/1025995/1. The work of the fourth author was partially supported by Simons Foundation grant 631302 and by NSF grant DMS-2054424. \dagger Department of Mathematics, University College London, 25 Gordon Street, London, WC1H 0AY, UK ([email protected]). \ddagger CNRS and Institut de Math\e'matiques de Toulouse, UMR5219; Universit\e' de Toulouse, CNRS; UPS, F-31062 Toulouse Cedex 9, France ([email protected]). \S Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, UK (E.A.Spence@ bath.ac.uk). \P Department of Mathematics, Northwestern University, Evanston, IL 60208-2730 USA (jwunsch@ math.northwestern.edu).
Keywords
- FEM
- Helmholtz
- hp-FEM
- splitting
ASJC Scopus subject areas
- Analysis
- Computational Mathematics
- Applied Mathematics