Noether type discrete conserved quantities arising from a finite element approximation of a variational problem

In this work we prove a weak Noether type theorem for a class of variational problems which include broken extremals. We then use this result to prove discrete Noether type conservation laws for certain classes of finite element discretisation of a model elliptic problem. In addition we study how well the finite element scheme satisfies the continuous conservation laws arising from the application of Noether's 1st Theorem (E. Noether 1918). We summarise extensive numerical tests, illustrating the conservativity of the discrete Noether law using the $p$--Laplacian as an example.Comment: 17 pages, 3 figures, 3 table

On discrete analogues of potential vorticity for variational shallow water systems

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.Comment: 12 pages, 1 figur

Moving Frames and Noether’s Conservation Laws – the General Case

In recent works [1, 2], the authors considered various Lagrangians, which are invariant under a Lie group action, in the case where the independent variables are themselves invariant. Using a moving frame for the Lie group action, they showed how to obtain the invariantized Euler-Lagrange equations and the space of conservation laws in terms of vectors of invariants and the adjoint representation of a moving frame. In this paper, we show how these calculations extend to the general case where the independent variables may participate in the action. We take for our main expository example the standard linear action of SL(2) on the two independent variables. This choice is motivated by applications to variational fluid problems which conserve potential vorticity. We also give the results for Lagrangians invariant under the standard linear action of SL(3) on the three independent variables

‘Field notes: contemporary art history as historiography’. Review of: Terry Smith, Art to Come: Histories of Contemporary Art, Durham and London: Duke University Press, 2019, 456 pp., 84 b. & w. illus., £92.00 hdbk, £25.99 pbk ISBN 9781478001942

Terry Smith characterizes Art to Come as a work of art historiography. The eleven chapters that comprise Art to Come–including several previously published essays by Smith–are primarily concerned with describing and analyzing art produced in the past few decades. This review takes up Smith’s invitation to understand Art to Come as historiography and argues that the book is a model for a mode of art writing that is simultaneously art historical and historiographical