A Generalized Fourier Transform for Boundary Element Methods with Symmetries∗

We study the solution of linear systems that typically arise in discretizations of boundary value problems on a domain with geometrical symmetries. If the discretization is done in an appropriate way, then such a system commutes with a group of permutation matrices. Recently, algorithms have been developed that exploit this special structure, however these methods are limited to the case that the permutations have no fixed points. Here a new symmetry exploiting algorithm, based on the Fourier transform on the symmetry group, is introduced which is capable of handling fixed points. The techniques developed can also be used to achieve further reductions when the right hand side of the proposed system has symmetries. The approach is illustrated by the boundary element method on an equilateral triangle and on a 3-cube. The reduction technique can also be applied to other solution methods, e.g., finite element methods.