Abstract
We study the resolvent for nontrapping obstacles on manifolds with Euclidean ends. It is well known that for such manifolds the outgoing resolvent satisfies (Formula Presented), but the constant C has been little studied. We show that, for high frequencies, the constant is bounded above by (Formula Presented) times the length of the longest generalized bicharacteristic of (Formula Presented) remaining in the support of x We show that this estimate is optimal in the case of manifolds without boundary. We then explore the implications of this result for the numerical analysis of the Helmholtz equation.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 157-202 |
| Number of pages | 46 |
| Journal | Pure and Applied Analysis |
| Volume | 2 |
| Issue number | 1 |
| DOIs | |
| State | Published - 2020 |
Funding
Galkowski was supported by NSF Postdoctoral Research Fellowship DMS-1502661 and thanks Maciej Zworski for helpful conversations. Spence was supported by EPSRC grant EP/R005591/1. Wunsch was partially supported by NSF grant DMS–1600023. The authors are grateful to an anonymous referee for helpful comments on the manuscript.
Keywords
- finite element method
- Helmholtz equation
- nontrapping
- resolvent
- variable wave speed
ASJC Scopus subject areas
- Analysis
- Mathematical Physics