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2020 Optimal constants in nontrapping resolvent estimates and applications in numerical analysis
Jeffrey Galkowski, Euan A. Spence, Jared Wunsch
Pure Appl. Anal. 2(1): 157-202 (2020). DOI: 10.2140/paa.2020.2.157

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 χR(k)χL2L2Ck1 for k>1, but the constant C has been little studied. We show that, for high frequencies, the constant is bounded above by 2π times the length of the longest generalized bicharacteristic of |ξ|g21 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.

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Jeffrey Galkowski. Euan A. Spence. Jared Wunsch. "Optimal constants in nontrapping resolvent estimates and applications in numerical analysis." Pure Appl. Anal. 2 (1) 157 - 202, 2020. https://doi.org/10.2140/paa.2020.2.157

Information

Received: 10 January 2019; Revised: 10 August 2019; Accepted: 26 October 2019; Published: 2020
First available in Project Euclid: 13 December 2019

zbMATH: 07159300
MathSciNet: MR4041281
Digital Object Identifier: 10.2140/paa.2020.2.157

Subjects:
Primary: 35J05 , 35P25 , 65N30

Keywords: finite element method , Helmholtz equation , nontrapping , resolvent , variable wave speed

Rights: Copyright © 2020 Mathematical Sciences Publishers

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