Running time is a key metric across the standard physical design flow stages. However, with the rapid growth in design sizes, routing runtime has become the runtime bottleneck in the physical design flow. To improve the effectiveness of the modern global router, we propose a global routing framework with GPU-accelerated routing algorithms and a heterogeneous task graph scheduler, called FastGR. Its runtime-oriented version FastGRL achieves 2.489× speedup compared with the state-of-the-art global router. Furthermore, the GPU-accelerated L-shape pattern routing used in FastGRL can contribute to 9.324× speedup over the sequential algorithm on CPU. Its quality-oriented version FastGRH offers further quality improvement over FastGRL with similar acceleration.