Three-dimensional numerical simulations of a four-wing flapping micro aerial vehicle (FMAV) with actual experimentally captured wing membrane kinematics have been performed using an immersed boundary method Navier-Stokes finite volume solver. To successfully simulate the clap and fling motion involving the wing intersection, the numerical solver has been specifically modified to use a newly improved interpolation template searching algorithm to prevent divergence. Reasonable agreement was found between the numerical and experimental results, with the first and second force peaks from the experimental results well captured by the simulations, which was not possible in the past. Moreover, a V-shaped linked vortex was observed, which was similar to the vortical structures found in other experiments and simulations. A wing drag analysis showed that the drag magnitude of the clap and fling configuration was about 2.5 times that of the single-wing configuration. Visualizations of the flowfields through pressure contours and vortical isosurfaces led to a better understanding of the underlying flapping-wing aerodynamics. The ability to accurately simulate the FMAV with flexible wings opened up many opportunities for further FMAV design-related problems.