Symmetrized Two-Scale Finite Element Discretizations for Partial Differential Equations with Symmetric Solutions

Abstract In this paper, some symmetrized two-scale finite element methods are proposed for a class of partial differential equations with symmetric solutions. With these methods, the finite element approximation on a fine tensor-product grid is reduced to the finite element approximations on a much coarser grid and a univariant fine grid. It is shown by both theory and numerics including electronic structure calculations that the resulting approximations still maintain an asymptotically optimal accuracy. By symmetrized two-scale finite element methods, the computational cost can be reduced further by a factor of 𝑑 approximately compared with two-scale finite element methods when Ω = ( 0 , 1 ) d \Omega=(0,1)^{d} . Consequently, symmetrized two-scale finite element methods reduce computational cost significantly.