Symmetrized two-scale finite element discretizations for partial differential equations with symmetric solutions

TL;DR

This paper introduces a symmetrized two-scale finite element method that efficiently approximates PDEs with symmetric solutions, maintaining accuracy while significantly reducing computational costs through a novel multi-grid approach.

Contribution

The paper presents a new symmetrized two-scale finite element method that reduces computational complexity for PDEs with symmetric solutions without sacrificing accuracy.

Findings

Maintains asymptotically optimal accuracy

Reduces computational cost significantly

Validated by theory and electronic structure numerics

Abstract

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.