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 domainwhere 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.

Numerical integration based on trivariate C 2 quartic spline quasi-interpolants

“…In order to compute (3.6), we recall that a trivariate polynomial p ∈ P 4 on a tetrahedron T of the partition T m can be represented in the Bernstein basis [3] as Since T is included in a cube with edge of length h, its volume is equal to h 3 24 and, since [3] …”

“…Concerning the numerical evaluation of 2D integrals, we mention the cubatures proposed in [8-10, 22, 30, 32], based on tensor product of univariate splines, on C 1 quadratic and C 2 quartic quasi-interpolating splines, defined on criss-cross triangulations and on Powell-Sabin partitions. Furthermore, numerical integration over polygons using an eight-node quadrilateral spline finite element is presented and studied in [23][24][25].…”