The Unit Circle Trap In Boundary Elements
Author(s)
G. S. Gipson & B. W. Yeigh
Abstract
Some of the traditional boundary element formulations suffer from instability
problems when a unit circle is involved in the discretization. This is a problem
that has been discussed frequently and handled empirically, but apparently never
formalized as a topic in the literature. This paper addresses how the problem
arises, identifies why the problem occurs, and demonstrates remedies to avoid it.
A theoretical outline will be provided and numerical case studies are presented.
Particular emphasis is placed on the Poisson equation and its synthesis.
Keywords: unit circle, Poisson equation, convergence, boundary elements,
numerical instability.
1 Introduction
This work focuses on a complaint that has been frequently discussed casually
among boundary element researchers but apparently has never been addressed
formally in the literature. The issue has to do with the unit circle which is typical
in normalized formulations of circular geometry. The purpose of this paper is to
first show one application in the Poisson equation where the problem arises and
to illustrate why it happens. The cause is identified and a solution is proposed.
This presentation is unusual in that it is best done using constant elements
since, as will be demonstrated, use of the more sophisticated, higher-order
elements exacerbate the problem. This lends credence to referring to the
problem as a \“trap.”
2 Background
The boundary element method is predicated upon defining a boundary value
problem in terms of equations involving surface integrals. Ideally, one would
Keywords
unit circle, Poisson equation, convergence, boundary elements,
numerical instability
Related Book
Other papers in this volume
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