Symmetry, groups, and boundary value problems. A progressive introduction to noncommutative harmonic analysis of partial differential equations in domains with geometrical symmetry
type: Item
instance of: scholarly article
title: Symmetry, groups, and boundary value problems. A progressive introduction to noncommutative harmonic analysis of partial differential equations in domains with geometrical symmetry
zbMATH Open document ID: 0578.73072
DOI: 10.1016/0045-7825(86)90119-2
published in: Computer Methods in Applied Mechanics and Engineering
publication date: 1986-01-01T00:00:00Z
review text: Let \(\Omega\) be a plane or spatial domain and G a group of isometries which leave \(\Omega\) invariant. Suppose one has to solve on \(\Omega\) a boundary value problem \(Au=f\) or an eigenvalue problem \(Au=\lambda u\). 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 f). 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 \(\Omega\). 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''. 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.
Mathematics Subject Classification ID: 74S30
Mathematics Subject Classification ID: 65T40
Mathematics Subject Classification ID: 74S05
Mathematics Subject Classification ID: 65M99
Mathematics Subject Classification ID: 65N99
Mathematics Subject Classification ID: 35J05
zbMATH DE Number: 3926398
zbMATH Keywords: plane or spatial domain
zbMATH Keywords: group of isometries
zbMATH Keywords: boundary value problem
zbMATH Keywords: eigenvalue problem
zbMATH Keywords: set of related problems
zbMATH Keywords: reduced domain
zbMATH Keywords: symmetry cell
zbMATH Keywords: n times smaller than the original
zbMATH Keywords: group representation
zbMATH Keywords: non- commutative harmonic analysis
author: Alain Bossavit
MaRDI profile type: Publication
full work available at URL: 0045-7825(86)90119-2
OpenAlex ID: W2041950884
cites work: Harmonic analysis as the exploitation of symmetry-a historical survey
cites work: Q3961021
cites work: Q3943153
cites work: Q5183974
cites work: Q5551264
cites work: Q4193645
cites work: Q3943552
cites work: Q5186261
cites work: Q5534201
cites work: Dynamics of rotationally periodic structures
cites work: Wave propagation and natural modes in periodic systems: II. Multi-coupled systems, with and without damping
cites work: Q5842186
cites work: Q3714586
cites work: Use of Group Theory in Various Integral Equations
cites work: Use of screw translational symmetry for the vibration analysis of structures
cites work: Analysis of symmetric or partially symmetric structures
cites work: Q3326480
cites work: Q3664082
cites work: Q4200225
cites work: On the possible application of group representations to some boundary value and eigenvalue problems of certain domains without classical symmetry. III: Some examples
description: scientific article; zbMATH DE number 3926398