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  -