Symmetry, groups, and boundary value problems. A progressive introduction to noncommutative harmonic analysis of partial differential equations in domains with geometrical symmetry

Abstract

Let Ω be a plane or spatial domain and G a group of isometries which leave Ω invariant. Suppose one has to solve on Ω a boundary value problem Au = f or an eigenvalue problem Au = μu. It is assumed (‘equivariance’) that the coefficients of A are also invariant by the action of G (but this is not required of the right-hand side ƒ). Then, instead of solving the original problem on the whole domain, one can solve a set of related problems (whose number does not exceed the order n of G) on a reduced domain, the ‘symmetry cell’, n times smaller than the original Ω. The process, which obviously promises interesting savings in structural analysis and other fields, generalizes Fourier analysis and may be referred to as “non-commutative harmonic analysis” [1]. The theoretical foundations are essentially those of group representation theory. The paper, which is mainly expository, aims at introducing the minimal amount of this theory necessary to understand how the process works and how it can be implemented in finite element codes. It also addresses the question (which seems to have been neglected in the literature) of the derivation of proper boundary conditions for the subproblems to be solved on the symmetry cell.