Numerical exploitation of equivariance

Linear operators in equations describing physical problems on a symmetric d omain often are also equivariant; which means that they commute with its sy mmetries, i.e., with the group of orthogonal transformations which leave th e domain invariant. Under suitable discretizations the resulting system mat rices are also equivariant with respect to a group of permutations. Methods for exploiting this equivariance in the numerical solution of linear syste ms of equations and eigenvalue problems via symmetry reduction are describe d. 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 s omewhat 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 he number of unknowns in the symmetry reduced system, the fatter does not but needs more overhead . Some examples are given to illustrate this situation. Our methods circumv ent 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 othe r purpose. A software package has been posted on the internet.