Standard Galerkin discretization techniques (with locally- or globally-supported basis functions) for boundary integral equations are inefficient for high frequency three dimensional exterior scattering simulations because they require a fixed number of unknowns per wavelength in each dimension, leading to large CPU time and memory requirements to set up the dense Galerkin matrix, with each entry requiring evaluation of multi-dimensional highly oscillatory integrals. In this work, using globally-supported basis functions, we describe an efficient fully discrete Galerkin surface integral equation algorithm for simulating high frequency acoustic scattering by three dimensional convex obstacles that includes a powerful integration scheme for evaluation of four dimensional Galerkin integrals with high-order accuracy. Such high-order order accuracy for various practically relevant frequencies (k∈[1,. 100,000]) substantially improves on approximations based on standard asymptotic techniques. We demonstrate the efficiency of our algorithm for spherical and non-spherical convex scattering for several wavenumbers 1≤k≤100,000 for low to high order prescribed tolerance. Our fully discrete algorithm requires only mild growth in the number of unknowns and CPU time as the frequency increases.