Numerical Exploitation of Equivariance

Abstract

Linear operators in equations describing physical problems on a symmetric domain often are also equivariant, which means that they commute with its symmetries, i.e., with the group of orthogonal transformations which leave the domain invariant. Under suitable discretizations the resulting system matrices are also equivariant with respect to a group of permutations. Methods for exploiting this equivariance in the numerical solution of linear systems of equations and eigenvalue problems via symmetry reduction are described. A very significant reduction in computational expense can be obtained in this way. The basic ideas underlying this method and its analysis involve group representation theory. The symmetry reduction method is complicated somewhat by the presence of nodes or elements which remain fixed under some of the symmetries. Two methods (regularization and projection) for handling such situations are described. The former increases the number of unknowns in the symmetry reduced system, the latter does not but needs more overhead. Some examples are given to illustrate this situation. Our methods circumvent the explicit use of symmetry adapted bases, but our methods can also be used to automatically generate such bases if they are needed for some other purpose. A software package has been posted on the internet.