Noether type discrete conserved quantities arising from a finite element approximation of a variational problem
This work provides a theoretical foundation for discrete conservation laws in finite element methods, which is relevant for numerical analysts working on structure-preserving discretizations.
The authors prove a weak Noether-type theorem for variational problems with broken extremals and derive discrete conservation laws for finite element discretizations of elliptic problems. Numerical tests with the p-Laplacian demonstrate the conservativity of the discrete Noether law.
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.