 
Summary: PIECEWISE POLYNOMIAL COLLOCATION FOR BOUNDARY
INTEGRAL EQUATIONS
KENDALL E. ATKINSON AND DAVID CHIEN y
Abstract. This paper considers the numerical solution of boundary integral equations of the
second kind, for Laplace's equation u = 0 on connected regions D in R3 with boundary S. The
boundary S is allowed to be smooth or piecewise smooth and we let f K j 1 K N g be a
triangulation of S. The numerical method is collocation with approximations which are piecewise
quadratic in the parametrization variables, leading to a numerical solution uN: Superconvergence
results for uN are given for S a smooth surface and for a special type of re nement strategy for the
triangulation. We show u ; uN is O( 4 log ) at the collocation node points, with the mesh size
for f Kg. Error analyses are given are given for other quantities and an important error analysis
is given for the approximation of S by piecewise quadratic interpolation on each triangular element,
with S either smooth or piecewise smooth. The convergence result we prove is only O( 2) but the
numericalexperimentssuggestthe resultis O( 4) for the errorat the collocationpoints, especiallyfor
S a smooth surface. The numericalintegrationof the collocationintegralsis discussed, and extended
numerical examples are given for problems involving both smooth and piecewise smooth surfaces.
Key words. Integral equations, quadrature interpolation, Laplace's equation, numerical inte
gration
AMS subject classi cations. 65R20, 35J05, 45L10, 65D05, 65D30
1. Introduction. In this work, we consider the numerical solution of boundary
