We outline how discrete analogues of the conservation of potential vorticity may be achieved in Finite Element numerical schemes for a variational system which has the particle relabelling symmetry, typically shallow water equations. We show that the discrete analogue of the conservation law for potential vorticity converges to the smooth law for potential vorticity, and moreover, for a strong solution, is the weak version of the potential vorticity law. This result rests on recent results by the author with T. Pryer concerning discrete analogues of conservation laws in Finite Element variational problems, together with an observation by P. Hydon concerning how the conservation of potential vorticity in smooth systems arises as a consequence of the linear momenta. The purpose of this paper is to provide all the necessary information for the implementation of the schemes and the necessary numerical tests. A brief tutorial on Noether's theorem is included to demonstrate the origin of the laws and to demonstrate that the numerical method follows the same basic principle, which is that the law follows directly from the Lie group invariance of the Lagrangian.

翻译：摘要：本文概述了如何在具有粒子重标对称性的变分系统（典型为浅水方程）的有限元数值格式中实现位势涡度守恒的离散类比。我们证明了位势涡度守恒律的离散类比收敛于光滑位势涡度律，并且对于强解而言，该离散形式是位势涡度律的弱形式。这一结论基于作者与T. Pryer近期关于有限元变分问题中守恒律离散类比的研究成果，以及P. Hydon关于光滑系统中位势涡度守恒如何由线性动量导出的观察。本文旨在提供实施这些格式所需的全部必要信息及数值测试。文中还包含关于诺特定理的简要教程，用以阐明守恒律的起源，并展示数值方法遵循相同的基本原理，即守恒律直接来源于拉格朗日量的李群不变性。