Symmetry-preserving finite element schemes: An introductory investigation
This work provides a method to incorporate symmetries into finite element schemes, potentially improving accuracy for problems with known symmetries, but it is an introductory investigation with limited scope.
The authors present a procedure for constructing symmetry-preserving finite element methods for second-order ODEs and extend it to (1+1)-D PDEs using Burgers' equation. Numerical simulations show that the invariant schemes converge at expected rates and outperform non-invariant counterparts.
Using the method of equivariant moving frames, we present a procedure for constructing symmetry-preserving finite element methods for second-order ordinary differential equations. Using the method of lines, we then indicate how our constructions can be extended to (1+1)-dimensional evolutionary partial differential equations, using Burgers' equation as an example. Numerical simulations verify that the symmetry-preserving finite element schemes constructed converge at the expected rate and that these schemes can yield better results than their non-invariant finite element counterparts.