Abstract
It is well known that when the geometry and/or coefficients allow stable trapped rays, the solution operator of the Helmholtz equation (a.k.a. the resolvent of the Laplacian) grows exponentially through a sequence of real frequencies tending to infinity. In this paper we show that, even in the presence of the strongestpossible trapping, if a set of frequencies of arbitrarily small measure is excluded, the Helmholtz solution operator grows at most polynomially as the frequency tends to infinity. One significant application of this result is in the convergence analysis of several numerical methods for solving the Helmholtz equation at high frequency that are based on a polynomialgrowth assumption on the solution operator (e.g. $hp$finite elements, $hp$boundary elements, certain multiscale methods). The result of this paper shows that this assumption holds, even in the presence of the strongestpossible trapping, for most frequencies.
Original language  English 

Journal  Communications on Pure and Applied Mathematics 
Early online date  31 Jul 2020 
DOIs  
Publication status  Epub ahead of print  31 Jul 2020 
Keywords
 math.AP
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Profiles

Euan Spence
 Department of Mathematical Sciences  Professor
 EPSRC Centre for Doctoral Training in Statistical Applied Mathematics (SAMBa)
Person: Research & Teaching