# American Institute of Mathematical Sciences

June  2006, 5(2): 337-347. doi: 10.3934/cpaa.2006.5.337

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

 1 Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL 60439, United States

Received  March 2005 Revised  May 2005 Published  March 2006

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.
Citation: J. N. Lyness. Extrapolation expansions for Hanging-Chad-Type Galerkin integrals with plane triangular elements. Communications on Pure & Applied Analysis, 2006, 5 (2) : 337-347. doi: 10.3934/cpaa.2006.5.337
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