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

TL;DR

This paper establishes a weak Noether type theorem for variational problems and demonstrates discrete conservation laws in finite element methods, supported by numerical tests on the p-Laplacian.

Contribution

It introduces a novel weak Noether theorem applicable to broken extremals and finite element discretizations, linking continuous and discrete conservation laws.

Findings

Finite element schemes approximately satisfy continuous conservation laws.

Numerical tests confirm the conservation properties using the p-Laplacian.

Discrete Noether laws demonstrate conservativity in numerical approximations.

Abstract

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.