We perform full elastic wave-field simulations within the Taipei basin by using a three-dimensional (3D) discontinuous finite-difference method. The 3D Taipei basin model is determined from a seismic reflection study. Two major subsurfaces, the Songshan formation (surface soil layer) and the basin basement, are constituted in the model. A parallel-based composite grid technique, a containing scalene grid and a discontinuous grid, is developed in this study to deal with the possible numerical problem of thin depth and low velocity of the Songshan formation. Taking advantage of the composite grid, the resolution of the subsurface structure can be reached to 20 m, and a higher frequency (up to 3 Hz) of the synthetic waveform can be achieved. In our strong ground motion simulations, we assume a constant velocity in each subsurface. Three different types of models are considered in the study: the Songshan formation with a basement structure model, a basin basement model, and a layered half-space model. Results indicate that only the model with both the Songshan formation and the basement structure can produce the apparent basin amplification effects. First, the surface wave generated after the primary S wave is trapped at the shallow part of the basin. Then, when the wave propagates through the deepest part of the basin, most of the energy is reflected from the boundary and focused back into the basin. In addition, part of the seismic wavefront turns and follows the shallow basin edge resulting in further amplification. Our study indicates that the complex Taipei basin geometry and fairly low velocity of the Songshan formation dominate the amplification and wave propagation behavior that result in extraordinary strong shaking patterns in the Taipei metropolitan region.