Numerical exploitation of symmetry in integral equations

Abstract

Linear integral operators describing physical problems on symmetric domains often are equivariant, which means that they commute with certain symmetries, i.e., with a group of orthogonal transformations leaving the domain invariant. Under suitable discretizations the resulting system matrices are also equivariant. A method for exploiting this equivariance in the numerical solution of linear equations and eigenvalue problems via symmetry reduction is described. A very significant reduction in the computational expense in both the assembling of the system matrix and in solving linear systems can be obtained in this way. This reduction is particularly important because the system matrices are typically full. The basic ideas underlying our method and its analysis involve group representation theory. We concentrate here on the use of symmetry adapted bases and their automated generation. In this paper symmetry reduction is studied in connection with quadrature formulae and the Nystrom method. A software package has been posted on the Internet.