Vol. 2, No. 1, 2020

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Optimal constants in nontrapping resolvent estimates and applications in numerical analysis

Jeffrey Galkowski, Euan A. Spence and Jared Wunsch

Vol. 2 (2020), No. 1, 157–202
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)χL2L2 Ck1 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|g2 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.

Keywords
resolvent, Helmholtz equation, nontrapping, variable wave speed, finite element method
Mathematical Subject Classification 2010
Primary: 35J05, 35P25, 65N30
Milestones
Received: 10 January 2019
Revised: 10 August 2019
Accepted: 26 October 2019
Published: 11 December 2019
Authors
Jeffrey Galkowski
Department of Mathematics
Northeastern University
Boston, MA
United States
Euan A. Spence
Department of Mathematical Sciences
University of Bath
Bath
United Kingdom
Jared Wunsch
Department of Mathematics
Northwestern University
Evanston, IL
United States