Numerical Exploitation of Equivariance4;5

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 dis-cretizations the resulting system matrices are also equivariant. Methods for exploiting this equivariance in the numerical solution of linear systems of equations and eigenvalue problems via symmetry reduction are described. A very signiicant 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 xed 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 they can be used to automatically generate them. A software package has been posted on the internet. 1. Introduction Many problems in science and mathematics exhibit symmetry phenomena which may be exploited to analyze them, and also to eeect a signiicant cost reduction in their numerical treatment. Usually the symmetry stems from the domain or body on which the problem is considered. The numerical treatment of problems such as partial diieren-tial equations and integral equations generally involves discretizations which ought (as far as possible) to incorporate or respect such symmetries. The present paper summarizes some of the recent work of the authors concerning systematic techniques for exploiting symmetry in the numerical treatment of systems of linear equations that arise from discretizing operator equations displaying symmetries. The unifying concept is a generalization of the Fourier transform for arbitrary nite groups. We study here the general case which incorporates non-abelian groups (i.e., having irreducible representations of dimension > 1) and the possibility that some nodes of the discretization remain xed under some of the symmetries. This latter case is of considerable practical importance since it naturally occurs in the most frequently used nite element or boundary element discretizations, and since it considerably complicates the algorithmic approach. We give here a uniied and simpliied view. The regularization techniques discussed in Section 6 are new. Although the algebraic tool employed here is the classical representation theory of groups, the …