Boundary Value Problems with Symmetry and Their Approximation by Finite Elements

Abstract

Symmetry, and related questions of group theory, especially representation theory, are well-known, popular topics among physicists. This paper describes how these classical ideas can be put to use in the realm of finite element computations by substituting a family of independent problems on a reduced domain (the “symmetry cell”) for the original problem on the whole domain. Two specific difficulties appear : one is how to set boundary conditions on symmetry planes (representation theory gives the answer); the other is how to proceed with the assembly of finite elements that constitute the symmetry cell. To deal with this, this paper introduces the concept of “index” of an element as part of the whole structure, which is the number, of its distinct symmetrical images. The rule for assembly then becomes to perform as usual, but to count each element as many times as its index.