Symmetry-preserving finite element schemes: An introductory investigation

TL;DR

This paper introduces a method to develop finite element schemes that preserve symmetries in differential equations, demonstrating improved accuracy and convergence through numerical simulations for both ODEs and PDEs.

Contribution

It presents a novel procedure using equivariant moving frames to construct symmetry-preserving finite element methods for second-order ODEs and extends this approach to PDEs like Burgers' equation.

Findings

Schemes converge at expected rates

Symmetry-preserving schemes outperform non-invariant ones

Method applicable to both ODEs and PDEs

Abstract

Using the method of equivariant moving frames, we present a procedure for constructing symmetry-preserving finite element methods for second-order ordinary differential equations. Using the method of lines, we then indicate how our constructions can be extended to (1+1)-dimensional evolutionary partial differential equations, using Burgers' equation as an example. Numerical simulations verify that the symmetry-preserving finite element schemes constructed converge at the expected rate and that these schemes can yield better results than their non-invariant finite element counterparts.