# On the development of symmetry-preserving finite element schemes for ordinary differential equations

## TL;DR

This paper presents a new finite element method for ordinary differential equations that preserves their Lie point symmetries, leading to improved long-term accuracy and applicability to various symmetry groups and polynomial degrees.

## Contribution

It introduces a symmetry-preserving finite element scheme using equivariant moving frames, applicable to diverse ODEs and polynomial degrees, with demonstrated numerical benefits.

## Key findings

- Symmetry-preserving schemes show better long-term accuracy.
- Applicable to arbitrary order ODEs and polynomial degrees.
- Numerical experiments confirm improved performance.

## Abstract

In this paper we introduce a procedure, based on the method of equivariant moving frames, for formulating continuous Galerkin finite element schemes that preserve the Lie point symmetries of initial value problems for ordinary differential equations. Our methodology applies to projectable and non-projectable symmetry group actions, to ordinary differential equations of arbitrary order, and finite element approximations of arbitrary polynomial degree. Several examples are included to illustrate various features of the symmetry-preserving process. We summarise extensive numerical experiments showing that symmetry-preserving finite element schemes may provide better long term accuracy than their non-invariant counterparts and can be implemented on larger elements.