\indent The continuum self-consistent field theory is applied to the study of the adsorption of flexible polyelectrolyte (PE) onto the surfaces of two two-dimensional charged square objects with a constant electric field strength immersed in a weakly charged polyelectrolyte solution. The dependences of the different chain conformations, i.e., bridging, loop, tail and train, and in particular, the bridging chain conformation, on various system parameters (the charge fraction of the PE chains, the surface charge density, the object size, the salt concentration, etc.) are investigated. The efficient Multigrid method is adopted to numerically solve the modified diffusion equation and the Poisson equation. It is found that, the thickness L{\scriptsize B} of the boundary layer of the adsorbed PE chains is independent of the chain length, and scales with the surface charge density $\sigma$ and the fraction of charges on PE chains $\alpha_{p}$ as L{\scriptsize B}$\sim\sigma ^{0.36}$ and L{\scriptsize B}$\sim\alpha _{p}^{0.36}$, respectively. Simulation results reveal that, the total amount of bridging chain conformation in the system scales linearly with respect the size of the charge objects, and scales linearly with the chain length in the long polymer chain regime. Simulation results reveal that, the total amount of the bridging chain conformation in the system scales with the charge fraction of PE chains as a power law, and the scaling exponent is dependent on all the other system parameters. Simulation results show that, the total amount of charges on the adsorbed chains in the system can over-compensate the surface charges for relatively long chains with high charge fractions.