Global polynomial approximation for Symm's equation on polygons

Abstract

Summary. To solve 1D linear integral equations on bounded intervals with nonsmooth input functions and solutions, we have recently proposed a quite general procedure, that is essentially based on the introduction of a nonlinear smoothing change of variable into the integral equation and on the approximation of the transformed solution by global algebraic polynomials. In particular, the new procedure has been applied to weakly singular equations of the second kind and to solve the generalized air foil equation for an airfoil with a flap. In these cases we have obtained arbitrarily high orders of convergence through the solution of very-well conditioned linear systems. In this paper, to enlarge the domain of applicability of our technique, we show how the above procedure can be successfully used also to solve the classical Symm's equation on a piecewise smooth curve. The collocation method we propose, applied to the transformed equation and based on Chebyshev polynomials of the first kind, has shown to be stable and convergent. A comparison with some recent numerical methods using splines or trigonometric polynomials shows that our method is highly competitive.