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.

翻译：在本文中,为具有对称解决方案的局部差分方程类别提议了一种对称的双尺度有限要素方法。 采用这种方法, 微强产品网格上的有限要素近似值将缩小为粗糙网格和单一微小网格上的有限要素近近似值。 包括电子结构计算在内的理论和数字都表明, 由此得出的近似值仍然保持了无损最佳的准确性。 因此, 平衡的双尺度有限要素方法极大地降低了计算成本 。