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

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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$P_{0}=(x_{0},y_{0})\in{\stackrel{\circ}{\Omega}}$\end{document}, 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.