Restriction matrices for numerically exploiting symmetry

In this paper we develop a technique for exploiting symmetry in the numerical treatment of boundary value problems (BVP) and eigenvalue problems which are invariant under a finite group G of congruences R^m. This technique will be based upon suitable restriction matrices strictly related to a system of irreducible matrix representation of G. Both Abelian and non-Abelian finite groups are considered. In the framework of symmetric Galerkin boundary element method (SGBEM), where the discretization matrices are typically full, to increase the computational gain we couple Panel Clustering Method [30] and Adaptive Cross Approximation algorithm [13] with restriction matrices introduced in this paper, showing some numerical examples. Applications of restriction matrices to SGBEM under the weaker assumption of partial geometrical symmetry, where the boundary has disconnected components, one of which is invariant, are proposed. The paper concludes with several numerical tests to demonstrate the effectiveness of the introduced technique in the numerical resolution of Dirichlet or Neumann invariant BVPs, in their differential or integral formulation.