New embedded boundary-type quadrature formulas for the simplex

Abstract

In this paper we consider the problem of the approximation of the integral of a smooth enough function f (x, y) on the standard simplex 2 ⊂ IR 2 by cubature formulas of the following kind:where the nodes (x α , y α ) , α = 1, 2, 3 are the vertices of the simplex. Such kind of quadratures belong to a more general class of formulas for numerical integration, which are called boundary-type quadrature formulas. We discuss three classes of such formulas that are exact for algebraic polynomials and generate embedded pairs. We give bounds for the truncation errors and conditions for convergence. Finally, we show how to organize an algorithm for the automatic computation of the quadratures with estimate of the errors and provide some numerical examples.Keywords Boundary type quadrature · Simplex · Algebraic degree of exactness Mathematics Subject Classifications (2000)Primary 65D30 · 65D32 · Secondary 65D05 · 65D15 To Prof. Walter Gautschi for his 50 years of professional activity.

On Some Numerical Integration Formulas on the d-Dimensional Simplex

“…To test the approximation accuracies of the proposed formulas, we consider the case d = 2 and the standard triangle S = Δ 2 of vertices v 0 = (0, 0), v 1 = (1, 0), v 2 = (0, 1). The numerical experiments are conducted by considering the following set of test functions [1] In all the experiments, the exact value of the integrals for functions f 1 and f 2 are computed by assuming as exact the numerical integration performed by Mathematica. In Table 1, we report the absolute value of the remainder terms E Δ2 r [f i ] = Q Δ2 r [f i ] − Δ2 f i (x)dx, i = 1, .…”

“…To reach our goal, we follow the approach proposed in Refs. [1,2]. More precisely, we approximate the integrand function f with a polynomial interpolant L S r [f ](x) which uses functional and derivative data values up to a fixed order r ∈ N at the vertices of S, i.e.…”

“…More precisely, we approximate the integrand function f with a polynomial interpolant L S r [f ](x) which uses functional and derivative data values up to a fixed order r ∈ N at the vertices of S, i.e. 1) and then, we integrate both sides of (1. The obtained quadrature formula (1.2) uses function and derivative data up to the order r at the vertices of S and has degree of exactness 1 + r, i.e.…”

The Complementary q-Lidstone Interpolating Polynomials and Applications

“…The analogous problem for the classical case was posed by Agarwal and Wong [12] and studied in [13,14]. Secondly, we are interested in applying such expansions to the construction of the boundary-type quadrature formula on triangles (see [15]) or to a solution of Hermite-Birkhoff interpolation problems on scattered data (see [16,17]).…”

Sobre una clase de fórmulas de cubicación encajadas en el simplex

“…In this series of numerical test the functions are the same used in [8]: the results are comparable.…”