On Some Numerical Integration Formulas on the d-Dimensional Simplex

In this paper, we consider the problem of the approximation of the integral of a function f over a d-dimensional simplex S of Rd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^{d}$$\end{document} by some quadrature formulas which use only the functional and derivative values of f on the boundary of the simplex S or function data at the vertices of S, at points on its facets and at its center of gravity. The quadrature formulas are computed by integrating over S a polynomial approximant of f which uses functional and derivative values at the vertices of S.


Introduction
The problem of the determination of quadrature rules for triangles, tetrahedra and, in general, for d-dimensional simplicial domains has reached the attention of a number of scholars starting from the middle of the nineteenth century up today (see [13] and the references therein). Although many papers focus on quadrature rules for triangles [3,10,12,16], only a limited literature is available on the integration in three or higher dimensions [4,14,17]. In this paper, we approach the problem of integration over general d -dimensional simplices by special type integration formulas which use functional and derivative values of the integrand function f mainly on points on the boundary of the d-dimensional simplex S. When the nodes lie only on the boundary of S these formulas are called boundary type quadrature formulas and are used when the values of f and its derivatives inside the simplex are not given or are not easily determinable. Applications of these formulas can be realized in the framework of the numerical solution of boundary value problems of partial differential equations (see [9] and the references therein).
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.
and then, we integrate both sides of (1.1) over the d-dimensional simplex S to get the quadrature formula 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. E S r [f ] = 0 whenever f is a polynomial in d variables of total degree at most 1 + r. The main feature of the quadrature formula Q S r [f ] is that it relies only on function and derivative data up to the order r at the vertices of S. This motivates us to look for approximations of those derivatives to obtain quadrature formulas which do not use any derivative data. To this end, we restrict to the case r = 1 and, by following the technique described in Ref. [6], we approximate the first order derivative data by three-point finite differences approximation. According to the choice of the approximation of the derivative data, we get different quadrature formulas with degree of exactness 2 and, to increase the degree of exactness of such formulas, we consider the convex combination of two of them to get a quadrature formula with degree of exactness 3 (see Sect. 3). Finally, we restrict to the two-dimensional case (see Sect. 4) and we provide numerical results to test the approximation accuracies of the proposed formulas (see Sect. 5).

Preliminaries and Notations
We also set λ = (λ 0 , λ 1 , λ 2 , . . . , λ d ) and λ l = (λ 0 , . . . , λ l−1 , λ l+1 , . . . , λ d ) for each l = 0, . . . , d. Moreover, we denote by the k-th order directional derivative of f along the line segment between x and v. Finally, we use the notations for the derivative of f along the directed line segment from x j to x i (as usual, · denotes the dot product) and for the composition of derivatives along the directed sides of the simplex. Under these assumptions, we get the following result.

5)
for any k ∈ N, k ≤ r and x ∈ R d .
Proof. By the equality (2.6) and by recalling that 0 ≤ λ j (x) ≤ 1 for each x ∈ S, we easily obtain and, by Proposition 2.2, we easily get the inequality (2.8).

Construction of the Quadrature Formula
The multivariate Lagrange interpolation polynomial on the simplex S in barycentric coordinates is The operator L S reproduces polynomials up to the degree 1 and interpolates the values of f at the vertices v l of the simplex S. If the function f belongs to C r (S), we can replace the values f (v l ) by the modified Taylor polynomial of degree r at v proposed in [6], the resulting polynomial operator is Remark 2.4. Since S is a compact convex domain and L S is a linear bounded operator, in line with [8], L S r can be interpreted as and, from [8,Proposition 3.4], it follows that L S r inherits the interpolation properties of the Lagrange operator (2.9).
To obtain the desired quadrature formula, we rearrange polynomial (2.10) by taking into account Lemma 2.1. More precisely, The quadrature formula is then computed by integrating the right hand side of (2.13) on the simplex S. and Moreover, the quadrature formula Q S r [f ] has degree of exactness 1 + r .
Proof. By integrating the right hand side of equality (2.13), we get (2.16) and then (2.16) becomes

Error Bounds
To give a bound for the remainder term E S r [f ] of the quadrature formula in Theorem 2.5, we need some additional notations. More precisely, for a k-times continuous differentiable function f : S → R, we introduce the norm where · denotes the Euclidean norm in R d , and y is assumed to be a column vector. Consequently, for any x ∈ S and y ∈ R d , we have Proof. By taking the modulus of both sides of equality (2.15), by applying the triangular inequality and by bounding the directional derivative of f of order r + 2 by (2.19), we have Using the inequality in Proposition (2.3), and by the fact that

Integration Formulas on the Simplex with Only Function Data
The main feature of the quadrature formula (2.14) is that it uses only derivatives of f along the edges of S; this motivates to consider approximations of those derivatives to obtain quadrature formulas which use the function data at the vertices of the simplex S, at points on its facets or at its center of gravity. To this aim, we focus on the case r = 1 and we consider different kinds of approximation of the derivatives in (2.14). For r = 1, the quadrature formula (2.14) in Theorem 2.5 becomes where the differences of directional derivatives along the edges of the simplex S can be replaced by a three-point finite difference approximation. To do this, let us recall that for a univariate function g, it is possible to consider the derivation formula for some ξ ∈ [a − h, a + h]. Using this formula with h = ±1/2 and a = 1/2 we get a three-point approximation for g (0) − g (1) with a remainder term which is expressed in terms of the modulus of continuity of g [6, Section 5.1]. By applying the formula (3.4) along the edges of S we get where ω denotes the modulus of continuity with respect to t ∈ [0, 1]. By substituting expression (3.5) in (3.3) and by rearranging, we get Finally, by substituting (3.6) in (3.1), we get . For all α ∈ (0, 1) we have
and, since λ k (g l ) = 1 d , for each l, k = 0, . . . , d, then By substituting (3.9) in (3.1), we get (3.10) To have a three point finite difference approximation of the directional derivatives in (3.10), for each l = 0, . . . , d, let us introduce the univariate function For t = 1 and t = α ∈ (0, 1), the second-order Taylor expansion of h l (1) and h l (α) centered at 0 with integral remainder are (3.12) and Then, by (3.12) and (3.13), we get where Therefore, by (3.14) it follows that (3.15) By rewriting equality (3.15) in terms of f we get where Finally, by substituting (3.16) in (3.1), we get (3.7). R(α)[f ] = 0 whenever f is a polynomial in d variables of total degree 2 and, therefore, the quadrature formula (3.7) has degree of exactness 2.
that is a quadrature formula which uses only the function data at the points g l and dg l +4v l d+4 , l = 0, . . . , d, and is exact for all polynomial of degree less than or equal to 2. 2. For α = d d+1 , the quadrature formula (3.7) becomes, where x * is the center of gravity of S.
To improve the approximation accuracy of the quadrature formula (3.19), let us consider a convex combination of this formula with a multivariate Simpson rule for a simplex proposed in [7, Theorem 5.1]. For a particular value of the parameter of the linear combination, we are able to get a quadrature formula with an higher degree of exactness. Then,

20)
where Proof. Equality (4.13) follows by setting α = 1 3 in (4.11). To prove the degree of exactness of the formula (4.12), we proceed by verifying the exactness of the quadrature formula for the monomials x 4 , x 3 y, xy 3 , x 2 y 2 , y 4 , similarly to the proof of Theorem 3.5 for d = 2.