Extrapolation expansions for Hanging-Chad-Type Galerkin integrals with plane triangular elements

Pages: 337 - 347, Volume 5, Issue 2, June 2006

doi:10.3934/cpaa.2006.5.337       Abstract        Full Text (195.7K)       Related Articles

J. N. Lyness - Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL 60439, United States (email)

Abstract: Applications of three-dimensional Galerkin boundary element methods require the numerical evaluation of many four-dimensional integrals. In this paper we explore the possibility of using extrapolation quadrature. To do so, one needs appropriate error functional expansions. The treatment here is limited to integration over a region $\mathcal T_1 \times \mathcal T_2$, where $\mathcal T_1$ and $\mathcal T_2$ are planar triangular elements in a hanging-chad configuration; that is, they have one vertex in common but are otherwise disjoint. We derive error expansions for product trapezoidal rules valid for integrands having an $|r_{12}|^{-1}$ factor. This factor gives rise to a weak singularity at the common vertex.

Keywords:  Galerkin integral, extrapolation, multidimensional quadrature, hanging-chad, singular integrals.
Mathematics Subject Classification:  Primary: 63D30, 65D32; Secondary: 65B05, 65N38.

Received: March 2005;      Revised: May 2005;      Published: March 2006.