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 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.

Invariant Variational Schemes for Ordinary Differential Equations

“…As the schemes developed in [23], the proposed methodology is variational, in that we discretize the Lagrangian rather than the associated Euler-Lagrange equations. Furthermore, as the schemes proposed in [4][5][6][7]10], our schemes are invariant as well, due to the well-known fact that symmetries of the Lagrangian are also symmetries of the corresponding Euler-Lagrange equations [25]. Lastly, similar to the exactly conservative schemes derived in [29,30], our schemes will also be exactly conservative, thanks to Noether's theorem.…”

“…We note that since the constraint ∆x 2 k + ∆u 2 k − is invariant under translations and rotations, the constrained Lagrangrian (33) is SE(2) invariant. After the multiplication by 5 , to avoid small denominators, the resulting Euler-Lagrange equations are…”

“…In doing so, geometric integrators typically provide better global and long term numerical results than comparable non-geometric methods. Typical examples include, amongst others, symplectic integrators, [3,15,18,27], Lie-Poison structure preserving schemes, [31], energy-preserving methods, [26], exactly conservative schemes, [29,30], symmetry-preserving methods, [4][5][6][7]10], and variational integrators, [23].…”

Invariant variational schemes for ordinary differential equations

“…The invariant variational approach outlined in the previous paragraph offers several advantages over other related geometric integrators. First, compared to invariant integrators, [9][10][11][12][13] that only focus on preserving the symmetries of the Euler-Lagrange equations, without consideration to its variational origin, the invariant variational schemes constructed in this paper have the additional benefit of preserving the conserved quantities of the problem. By preserving first integrals, the schemes should be more stable and produce better long-term numerical results, which is one of the main appealing properties of geometric numerical integrators.…”

“…In doing so, geometric integrators typically provide better global and long-term numerical results than comparable nongeometric methods. Typical examples include, among others, symplectic integrators, [1][2][3][4] Lie-Poison structure preserving schemes, 5 energy-preserving methods, 6 exactly conservative schemes, 7,8 symmetry-preserving methods, [9][10][11][12][13] and variational integrators. 14 In this paper, we use the method of moving frames, [15][16][17] to construct numerical schemes for ordinary differential equations that preserve variational symmetries of Euler-Lagrange equations.…”