Finite-part integrals over polygons by an 8-node quadrilateral spline finite element

Abstract

In this paper we consider the numerical integration on a polygonal domain Ω in ℝ2 of a function F(x,y) which is integrable except at a point \(P_{0}=(x_{0},y_{0})\in{\stackrel{\circ}{\Omega}}\), where F becomes infinite of order two. We approximate either the finite-part or the two-dimensional Cauchy principal value of the integral by using a spline finite element method combined with a subdivision technique also of adaptive type. We prove the convergence of the obtained sequence of cubatures. Finally, to illustrate the behaviour of the proposed method, we present some numerical examples.