Piecewise Polynomial Collocation for Integral Equations with a Smooth Kernel on Surfaces in Three Dimensions

Abstract

We consider solving integral equations on a piecewise smooth surface S in R 3 with a smooth kernel function, using piecewise polynomial collocation interpolation of the surface. The theoretical analysis includes the effects of the numerical integration of the collocation coefficients and the use of the approximating surface. The resulting order of convergence is higher than had previously been expected in the literature. Introduction. Consider the integral equation( 1)with k(P, Q) continuous for P, Q ∈ S, and with S a piecewise smooth surface in R 3 . We write the equation ( 1) assymbolically. We assume λ is nonzero and is not an eigenvalue of the integral operator K defined implicitly in (1). Thus, (1) has a unique solution f ∈ C(S) for each g ∈ C(S). In this paper we use collocation with piecewise quadratic interpolation for both the surface S and the unknown function f , as proposed in Atkinson [3].In practice, most of the 3-D boundary integral equations that arise do not have a smooth kernel. The major motivation of this paper is to develop the tools needed for handling boundary integral equations. Also, this paper is the first paper of a sequence of two papers. The second paper, Atkinson and Chien [6], will discuss a nonsmooth kernel case.

Two-Dimensional Quadrature for Functions with a Point Singularity on a Triangular Region

“…The function f is smooth on B0, and B0 is uniformly divided by triangular elements. By the results in [6], (4.18) follows.…”

“…The analysis given in [6] indicates that a quasi-uniform refinement is a better scheme to use with smooth integrands.…”

“…Since f is smooth on AK and rnK cr A/ is also smooth, we have onto (4.5) 6 f(mK(s, t)) E f(mK(pj))lj(s, t) Hf,K(S, t; , j=l where Hf,K(S, t; , rl) S-s + t-f(mi((, )) -s +t f(mt(,V))l(s,t)…”

“…We now have for Hf,1 and Hf,2 that the coefficient of s 3 in Hf,1 is (4.13) where 6t is the diameter of A1 and A2. This argument is based on Chien [6].…”

“…The empirical orders of convergence pn approach 1 and 2, respectively, as expected. 6. Generalization.…”