Groups of congruences and restriction matrices
The aim of the paper is to present a technique for exploiting symmetry in the numerical treatment of boundary value problems via finite difference, finite element and boundary element methods. This technique is based upon suitable restriction matrices strictly related to the mesh defined in the domain of the problem. The first section makes the connection between the groups of congruences and restriction matrices, and the practical problems arising from physics and engineering domains, in which one deals with structures having various geometrical symmetries. It is known that the use of discretizations of partial differential equations and boundary integral equations, which are suitably adapted to respect symmetry properties, yields useful decompositions, which can mainly reduce the computational cost and the memory storage. The second section recalls fundamental results concerning the theory of group representation. In the sections three and four one introduces the \(M\)-elementary restriction matrices related to a group \(G\) and to a system of irreductible matrix representations of \(G\), where \(G\) is a finite group of congruences which sents a domain \(\Omega\subset \mathbb{R}^m\) into itself. The number \(M\) depends only on the group \(G\). In this section the elementary restriction matrices corresponding to some simple groups of congruences and to their system of representations like the Abelian group generated by the symmetries of \(\mathbb{R}^2\) with respect to coordinate axes, the dihedral group of orthogonal matrices which preserve a square etc. are also constructed. In the fifth section, starting from these elementary restriction matrices, one constructs restriction matrices for any finite-dimensional space \(v(\Omega)\) of real functions defined on \((\Omega)\), when \(v(\Omega\)) is invariant with respect to \(G\). Restriction matrices for the space \(v(\Omega)\) have a block structure and all blocks can be obtained as from an elementary restriction matrix. Section six deals with the restriction matrices relative to a mesh. An algorithm is presented which is used for an example of construction of restriction matrices for a particular domain suitably discretized, given by an equilateral triangle centered in the origin of a Cartesian system and symmetric with respect to the axis \(x_1= 0\), with a hexagonal hole in its center. In the final section, one proposes a result of decomposition for linear systems with \(N\) equations and \(N\) unknowns using suitable restriction matrices. One proves that linear systems coming from the discretization of boundary value problems for elliptic differential operators invariant with respect to \(G\) are decomposable.
scientific article; zbMATH DE number 2057837