Piecewise Polynomial Collocation For Integral Equations On Surfaces In Three Dimensions

Of a thesis submitted in partial fulfillment of the requirements for the Doctor of Philosophy degree in Mathematics in the Graduate College of The University of Iowa May 1991 Thesis supervisor: Professor Kendall E. Atkinson 1 ABSTRACT We consider solving integral equations on a piecewise smooth surface S in R 3 with a smooth kernel using collocation with isoparametric piecewise polynomial interpolation. Symbolically, we write the integral equation as ( \Gamma K)f = g. We approximate both the surface S and the unknown solution f by using such interpolation. P n is the bounded projection such that P n f is the piecewise quadratic interpolant of f . For the collocation method, kf \Gamma f n k1 B kf \Gamma P n fk1 for an appropriate B and all sufficiently large n. We can define K n f by using piecewise quadratic interpolation of f and the surface S; and by the theory of the Nystrom method, we obtain a solution ~ f n , ( \Gamma K n ) ~ f n = g. The result of the two methods coinci..