  In this paper, a symmetrized two-scale finite element method is proposed for
a class of partial differential equations with symmetric solutions. With this
method, the finite element approximation on a fine tensor product grid is
reduced to the finite element approximations on a much coarse grid and a
univariant fine grid. It is shown by both theory and numerics including
electronic structure calculations that the resulting approximation still
maintains an asymptotically optimal accuracy. Consequently the symmetrized
two-scale finite element method reduces computational cost significantly.
