Generalized Duffy transformation for integrating vertex singularities

摘要

For an integrand with a 1/r vertex singularity, the Duffy transformation from a triangle (pyramid) to a square (cube) provides an accurate and efficient technique to evaluate the integral. In this paper, we generalize the Duffy transformation to power singularities of the form p(x)/r (alpha) , where p is a trivariate polynomial and alpha > 0 is the strength of the singularity. We use the map (u, v, w) -> (x, y, z) : x = u (beta) , y = x v, z = x w, and judiciously choose beta to accurately estimate the integral. For alpha = 1, the Duffy transformation (beta = 1) is optimal, whereas if alpha not equal 1, we show that there are other values of beta that prove to be substantially better. Numerical tests in two and three dimensions are presented that reveal the improved accuracy of the new transformation. Higher-order partition of unity finite element solutions for the Laplace equation with a derivative singularity at a re-entrant corner are presented to demonstrate the benefits of using the generalized Duffy transformation.