  This paper presents a multiscale Petrov-Galerkin finite element method for
time-harmonic acoustic scattering problems with heterogeneous coefficients in
the high-frequency regime. We show that the method is pollution- free also in
the case of heterogeneous media provided that the stability bound of the
continuous problem grows at most polynomially with the wave number k. By
generalizing classical estimates of [Melenk, Ph.D. Thesis 1995] and [Hetmaniuk,
Commun. Math. Sci. 5 (2007)] for homogeneous medium, we show that this
assumption of polynomially wave number growth holds true for a particular class
of smooth heterogeneous material coefficients. Further, we present numerical
examples to verify our stability estimates and implement an example in the
wider class of discontinuous coefficients to show computational applicability
beyond our limited class of coefficients.
