Experiments of magnetic reconnection driven by the laser in laboratory with the Helmholtz capacitor-coil target have been carried out for years. In the experiment, the reconnecting magnetic fields are induced by the electric current inside two conjugate coils. The electric current results from the flow of free electrons between two metal plates connecting to one another via two coils, and free electrons are produced as one of the metal plate is irradiated by the intense laser. This kind of experiments of reconnection differs from others that perform the reconnection process with the magnetic field produced directly by the Biermann battery effect. The three-dimensional numerical experiments in the magnetohydrodynamical framework to show for the first time how the reconnection process is driven by the laser with the Helmholtz capacitor-coil target. The numerical simulation results indicate clearly that the outflow plasma of magnetic reconnection shows characteristic accumulation around the coils. In addition, the reconnecting magnetic field produced by the induced electric current inside the coils could be as strong as 100 T, and the plasma beta (the ratio of the thermal energy density to the magnetic energy density) around the reconnection region is around 10^-2. Comparisons with the existing results of the laboratory experiments suggest that the numerical experiment duplicate most features of the laboratory experiments, which helps examine the physics behind the phenomena shown by the laboratory experiments.