A Note on Symplectic, Multisymplectic Scheme in Finite Element Method

TL;DR

This paper demonstrates that finite element methods can preserve symplectic and multisymplectic structures in one- and two-dimensional cases, explaining their practical accuracy.

Contribution

It reveals the intrinsic preservation of symplectic structures in finite element schemes, providing theoretical insight into their accuracy.

Findings

Finite element schemes preserve symplectic structures in 1D.

Finite element schemes preserve multisymplectic structures in 2D.

Preservation explains the practical accuracy of these algorithms.

Abstract

We find that with uniform mesh, the numerical schemes derived from finite element method can keep a preserved symplectic structure in one-dimensional case and a preserved multisymplectic structure in two-dimentional case in certain discrete version respectively. These results are in fact the intrinsic reason that the numerical experiments indicate that such finite element algorithms are accurate in practice.