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

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 ordinary differential equations. Our methodology applies to projectable and non-projectable actions for ordinary differential equations of arbitrary order, and interpolating functions of arbitrary 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.