Wavenumber-explicit convergence of the hp-FEM for the full-space heterogeneous Helmholtz equation with smooth coefficients

A convergence theory for the hp-FEM applied to a variety of constant-coefficient Helmholtz problems was pioneered in papers [35], [36], [15], [34]. This shows that, if solution operator is bounded polynomially wavenumber k, then Galerkin method quasioptimal provided that hk/p?C1 and p?C2log?k, where C1 sufficiently small, C2 large, both are independent k,h, p. The significance this result hk/p=C1 p=C2log?k, quasioptimality achieved with total number degrees freedom proportional kd; i.e., does not suffer from pollution effect. paper proves analogous heterogeneous (i.e. variable-coefficient) equation, posed Rd, d=2,3, Sommerfeld radiation condition at infinity, C? coefficients. We also prove bound on relative error particular case plane-wave scattering problem. These first ever results wavenumber-explicit equation variable