TY - JOUR N1 - This version is the author accepted manuscript. For information on re-use, please refer to the publisher?s terms and conditions. KW - Helmholtz equation KW - high frequency KW - perfectly-matched layer KW - pollution effect KW - finite element method KW - error estimate KW - semiclassical analysis TI - The hp-FEM applied to the Helmholtz equation with PML truncation does not suffer from the pollution effect AV - public IS - 7 EP - 1816 VL - 22 SP - 1761 JF - Communications in Mathematical Sciences PB - International Press of Boston, Inc. A1 - Galkowski, J A1 - Lafontaine, D A1 - Spence, E A A1 - Wunsch, J Y1 - 2024/// N2 - We consider approximation of the variable-coefficient Helmholtz equation in the exterior of a Dirichlet obstacle using perfectly-matched-layer (PML) truncation; it is well known that this approximation is exponentially accurate in the PML width and the scaling angle, and the approximation was recently proved to be exponentially accurate in the wavenumber k in [28]. We show that the hp-FEM applied to this problem does not suffer from the pollution effect, in that there exist C1, C2 > 0 such that if hk/p ? C1 and p ? C2 log k then the Galerkin solutions are quasioptimal (with constant independent of k), under the following two conditions (i) the solution operator of the original Helmholtz problem is polynomially bounded in k (which occurs for ?most? k by [41]), and (ii) either there is no obstacle and the coefficients are smooth or the obstacle is analytic and the coefficients are analytic in a neighbourhood of the obstacle and smooth elsewhere. This hp-FEM result is obtained via a decomposition of the PML solution into ?high-? and ?low-frequency? components, analogous to the decomposition for the original Helmholtz solution recently proved in [29]. The decomposition is obtained using tools from semiclassical analysis (i.e., the PDE techniques specifically designed for studying Helmholtz problems with large k). UR - https://dx.doi.org/10.4310/CMS.240918021620 ID - discovery10185834 ER -