Galkowski, J;
Spence, EA;
Wunsch, J;
(2020)
Optimal constants in nontrapping resolvent estimates and applications in numerical analysis.
Pure and Applied Analysis
, 2
(1)
pp. 157-202.
10.2140/paa.2020.2.157.
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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 ∥ ∥ χ R ( k ) χ ∥ L 2 → L 2 ≤ C k − 1 for k > 1 , but the constant C has been little studied. We show that, for high frequencies, the constant is bounded above by 2 π times the length of the longest generalized bicharacteristic of ∣ ∣ ξ ∣ ∣ 2 g − 1 remaining in the support of χ . 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.
| Type: | Article |
|---|---|
| Title: | Optimal constants in nontrapping resolvent estimates and applications in numerical analysis |
| Open access status: | An open access version is available from UCL Discovery |
| DOI: | 10.2140/paa.2020.2.157 |
| Publisher version: | http://dx.doi.org/10.2140/paa.2020.2.157 |
| Language: | English |
| Additional information: | This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions. |
| Keywords: | resolvent, Helmholtz equation, nontrapping, variable wave speed, finite element method |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics |
| URI: | https://discovery.ucl.ac.uk/id/eprint/10138604 |
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