Exploiting symmetry in boundary element methods

@article{Allgower1992ExploitingSI,
  title={Exploiting symmetry in boundary element methods},
  author={Eugene L. Allgower and Klaus B{\"o}hmer and Kurt Georg and Rick Miranda},
  journal={SIAM Journal on Numerical Analysis},
  year={1992},
  volume={29},
  pages={534-552}
}
Linear operator equations $\mathcal {L}f = g$ are considered in the context of boundary element methods, where the operator $\mathcal {L}$ is equivariant, i.e., commutes with the actions of a given finite symmetry group. By introducing a generalization of Reynolds projectors, a decomposition of the identity operator is constructed, which in turn allows the decomposition of the problem $\mathcal {L}f = g$ into a finite number of symmetric subproblems. The data function g does not need to possess… 
View via Publisher
50 Citations
Highly Influential Citations
2
Background Citations
11
Methods Citations
4

Tables from this paper

50 Citations

Restriction matrices for numerically exploiting symmetry
TLDR
A technique for exploiting symmetry in the numerical treatment of boundary value problems (BVP) and eigenvalue problems which are invariant under a finite group $\mathcal{G}$ of congruences of ${\rm{I\!R}}^{m}$.
Numerical Exploitation of Equivariance 4;5
TLDR
The present paper summarizes some of the recent work of the authors concerning systematic techniques for exploiting symmetry in the numerical treatment of systems of linear equations that arise from discretizing operator equations displaying symmetries.
A GENERALIZED FOURIER TRANSFORM FOR BOUNDARY ELEMENT METHODS WITH SYMMETRIES
TLDR
A new symmetry exploiting algorithm, based on the Fourier transform on the symmetry group, is introduced which is capable of handling fixed points and can also be used to achieve further reductions when the right hand side of the proposed system has symmetries.
Numerical exploitation of symmetricstructures in BEM 1
TLDR
The eecient solution of linear systems of equations and of eigenvalue problems with the Fourier transform method is discussed and some examples involving symmetric domains in the plane and in 3-space are concluded.
Numerical exploitation of symmetry in integral equations
TLDR
In this paper symmetry reduction is studied in connection with quadrature formulae and the Nyström method on the use of symmetry adapted bases and their automated generation.
Exploiting Symmetry in 3 D Boundary Element Methods
Many linear operator equations are defined on regions which are invariant under a group Γ of symmetry transformations. If in addition, the linear operator is equivariant with respect to Γ and if a…
Exploiting Symmetry in
: Classical integral operators usually display invariance with respect to orthogonal transformations. If the domain of an operator equation is symmetric with respect to some orthogonal…
Exploiting Symmetry in BEM 2
TLDR
The present paper gives a new uniied presentation (giving several modiied deenitions of the Fourier transform) of adapted to the situation and special needs of BEM, and illustrates the reductions which take place when the rich symmetries of a three dimensional cube are exploited.
On group Fourier analysis and symmetry preserving discretizations of PDEs
TLDR
Algorithms for computing matrix exponentials based on Fourier transforms on finite groups and novel applications of multivariate non-separable Chebyshev polynomials in the construction of spectral element bases on triangular and simplicial sub-domains are introduced.
Restrictions matrices for Platonic solids invariance and applications to space-time energetic BEM
...
...

References

SHOWING 1-10 OF 14 REFERENCES
A Survey of Boundary Integral Equation Methods for the Numerical Solution of Laplace’s Equation in Three Dimensions
The principal reformulations of Laplace’s equation as boundary integral equations (BIEs) are described, together with results on their solvability and the regularity of their solutions. The numerical…
An algorithm for piecewise lin-ear approximation of implicitly de?ned two-dimensional surfaces
Let $H:\mathbb{R}^{N + 2} \to \mathbb{R}^N $ be a smooth map and let $x_0 \in H^{ - 1} (0)$ be a regular point of H. We present an algorithm for obtaining a global piecewise linear approximation to a…
Computational methods for bifurcation problems with symmetries—with special attention to steady state and Hopf bifurcation points
Numerical continuation methods - an introduction
TLDR
This book discusses Continuation Methods, Newton's Method and Orthogonal Decompositions Revisited, and Update Methods and their Numerical Stability.
The Computation of Fixed Points and Applications
I Brouwer's Theorem.- II Some Applications of Brouwer's Theorem.- III Triangulations.- IV Algorithms to Find Completely-Labelled Simplices.- V Extensions of Brouwer's Theorem.- VI Applications of…
The domain reduction method: High way reduction in three dimensions and convergence with inexact sol
An independent suspension for the front wheels of motor vehicles by means of two guide members disposed one above the other whose pivot axes intersect in the center position of the wheel, on the one…
Singularities and groups in bifurcation theory
This book has been written in a frankly partisian spirit-we believe that singularity theory offers an extremely useful approach to bifurcation prob- lems and we hope to convert the reader to this…
A generalized equibranching lemma with applications to D4 × Z2 symmetric elliptic problems
  • Part I. Preprint, University of Marburg, Fed. Rep. Germany, submitted
  • 1990
A generalized equibranching lemma with applications to D 4 × Z 2 symmetric elliptic problems. Part I. Preprint
  • A generalized equibranching lemma with applications to D 4 × Z 2 symmetric elliptic problems. Part I. Preprint
  • 1990
Gruppentheorethische Methoden und ihre Anwendung, Teubner
  • Gruppentheorethische Methoden und ihre Anwendung, Teubner
  • 1979
...
...