Symmetry-Preserving Finite Element Schemes: An Introductory Investigation

Abstract

Using the method of equivariant moving frames, we present a procedure for constructing symmetrypreserving 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.

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

“…Recently, the first and third authors began to adapt some of the techniques used to construct symmetry-preserving finite difference schemes to finite element methods, [8], focusing on second order ODEs and Burgers' equation. From a numerical perspective, finite element methods offer several advantages over finite difference methods.…”

“…In [8], the authors limited themselves to second order ODEs and only considered projectable group actions. In the present paper we generalise the ideas set out in [8] to ODEs of arbitrary orders and to general Lie point transformation groups.…”

“…In [8], the authors limited themselves to second order ODEs and only considered projectable group actions. In the present paper we generalise the ideas set out in [8] to ODEs of arbitrary orders and to general Lie point transformation groups. In particular, we utilise the conforming nature of the finite element approximation to exploit the invariantisation procedure within a continuous framework, whereas in the prequel a finite difference like approach was considered.…”

“…In particular, we utilise the conforming nature of the finite element approximation to exploit the invariantisation procedure within a continuous framework, whereas in the prequel a finite difference like approach was considered. We note that viewing high degree finite elements in the difference setting leads to complex calculations, which adds computational challenges to the methodology considered in [8]. The approach introduced in this paper has several benefits, not least of which is the ability to construct invariant finite element approximations of arbitrarily high order simultaneously.…”

“…Though, we note that for some group actions it will be necessary to consider certain aspects of the invariantisation procedure for each polynomial degree independently. In these cases, these degree specific modifications are significantly simpler than those introduced in [8].…”