Restrictions matrices for Platonic solids invariance and applications to space–time energetic BEM

Abstract

Geometrical symmetry is found in many linear field problems. Group representation theory is the only valuable tool for exploiting this property from a computational point of view. In this context, here we apply a technique for taking into account equivariance in the numerical treatment of space–time boundary integral equations, which are invariant under a finite group G of congruences of R 3 , each related to one of the so-called Platonic solids. This technique is based upon suitable restriction matrices strictly related to a system of unitary, irreducible, pairwise not-equivalent matrix representations of G . The main development is expounded in the framework of space–time energetic boundary element method applied to Neumann exterior wave propagation problems, where the discretization matrices have a block lower triangular Toeplitz structure, and the diagonal block, to be inverted at each time step, is typically dense. Several numerical results, related to computational saving, are presented.