It has been demonstrated that circular dichroism (CD) signals from chiral molecules can be boosted by plasmonic nanostructures inducing strong local electromagnetic fields. To optimize nanostructures to improve CD enhancement, numerical simulations such as the finite element method (FEM) have been widely adopted. However, FEM calculations for CD have been frequently hampered by unwanted numerical artifacts due to improperly discretizing problem spaces. Here, we introduce a new meshing rule for FEM that provides CD simulations with superior numerical accuracy. We show that unwanted numerical artifacts can be suppressed by implementing the mirror-symmetric mesh configuration that generates identical numerical artifacts in the two-opposite circularly polarized waves, which cancel each other out in the final CD result. By applying our meshing scheme, we demonstrate a nanostructure/chiral molecule coupled system from which the CD signal is significantly enhanced. Since our meshing scheme addresses the previously unresolved issue of discriminating between very small CD signals and numerical errors, it can be directly applied to numerical simulations featuring natural chiral molecules which have intrinsically weak chiroptical responses.