on some numerical integration formulas on the d ......d α l = d 1 0,l d 2 1...d l − d αl+1...

Mediterr. J. Math. (2020) 17:142https://doi.org/10.1007/s00009-020-01579-31660-5446/20/050001-19published onlineAugust 20, 2020c© The Author(s) 2020

On Some Numerical Integration Formulason the d-Dimensional Simplex

Filomena Di Tommaso and Benaissa Zerroudi

Abstract. In this paper, we consider the problem of the approximationof the integral of a function f over a d-dimensional simplex S of Rd bysome quadrature formulas which use only the functional and derivativevalues of f on the boundary of the simplex S or function data at thevertices of S, at points on its facets and at its center of gravity. Thequadrature formulas are computed by integrating over S a polynomialapproximant of f which uses functional and derivative values at thevertices of S.

Mathematics Subject Classification. Primary 65D32, Secondary 65D30.

Keywords. Numerical integration, Cubature, Simplex.

1. Introduction

The problem of the determination of quadrature rules for triangles, tetra-hedra and, in general, for d-dimensional simplicial domains has reached theattention of a number of scholars starting from the middle of the nineteenthcentury up today (see [13] and the references therein). Although many papersfocus on quadrature rules for triangles [3,10,12,16], only a limited literatureis available on the integration in three or higher dimensions [4,14,17]. In thispaper, we approach the problem of integration over general d -dimensionalsimplices by special type integration formulas which use functional and de-rivative values of the integrand function f mainly on points on the boundaryof the d-dimensional simplex S. When the nodes lie only on the boundary ofS these formulas are called boundary type quadrature formulas and are usedwhen the values of f and its derivatives inside the simplex are not given orare not easily determinable. Applications of these formulas can be realized

This research has been accomplished within the RITA “Research ITalian network on Ap-proximation”. This research was supported by GNCS–INdAM 2019 project. The authorswould like to thank Professor F. Dell’Accio, Professor O. Nouisser and the anonymousreferee for their valuable comments.

http://crossmark.crossref.org/dialog/?doi=10.1007/s00009-020-01579-3&domain=pdf

http://orcid.org/0000-0002-4638-2994

142 of 19 F. Di Tommaso, B. Zerroudi MJOM

in the framework of the numerical solution of boundary value problems ofpartial differential equations (see [9] and the references therein).

To reach our goal, we follow the approach proposed in Refs. [1,2]. Moreprecisely, we approximate the integrand function f with a polynomial inter-polant LS

r [f ](x) which uses functional and derivative data values up to afixed order r ∈ N at the vertices of S, i.e.

f(x) = LSr [f ](x) + RS

r [f ](x), x ∈ S, (1.1)

and then, we integrate both sides of (1.1) over the d-dimensional simplex Sto get the quadrature formula∫

S

f(x)dx = QSr [f ] + ES

r [f ] (1.2)

where

QSr [f ] =

∫S

LSr [f ](x)dx and ES

r [f ] =∫

S

RSr [f ](x)dx.

The obtained quadrature formula (1.2) uses function and derivative dataup to the order r at the vertices of S and has degree of exactness 1 + r, i.e.ES

r [f ] = 0 whenever f is a polynomial in d variables of total degree at most1 + r. The main feature of the quadrature formula QS

r [f ] is that it reliesonly 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 obtainquadrature formulas which do not use any derivative data. To this end, werestrict 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 dif-ferences approximation. According to the choice of the approximation of thederivative data, we get different quadrature formulas with degree of exact-ness 2 and, to increase the degree of exactness of such formulas, we considerthe convex combination of two of them to get a quadrature formula with de-gree of exactness 3 (see Sect. 3). Finally, we restrict to the two-dimensionalcase (see Sect. 4) and we provide numerical results to test the approximationaccuracies of the proposed formulas (see Sect. 5).

2. A Quadrature Formula on the d-Dimensional Simplex

2.1. Preliminaries and Notations

Let S ⊂ Rd be a not degenerated d-dimensional simplex with vertices v0, . . . ,

vd ∈ Rd and

A(S) =

∣∣∣∣∣∣∣∣∣

1 v0

1 v1

1...

1 vd

∣∣∣∣∣∣∣∣∣the signed volume of the hyperparallelepiped with vertices v0, . . . ,vd. For apoint x ∈ S and for each l = 0, 1, . . . , d we denote by Sl(x) the d-dimensional

MJOM On Some Numerical Integration Formulas on the. . . of 19 142

simplex of vertices v0, . . . ,vl−1,x,vl+1, . . . ,vd. The barycentric coordinatesof x with respect to the simplex S are then defined by

λl(x) =A(Sl)A(S)

, l = 0, 1, . . . , d. (2.1)

For each α = (α1, α2, . . . , αd) ∈ Nd and x = (x1, . . . , xd) ∈ R

d, as usual, wedenote by

|α| = α1 + α2 + · · · + αd, α! = α1! . . . αd!, xα = xα11 xα2

2 . . . xαd

d .

We also set λ = (λ0, λ1, λ2, . . . , λd) and λl = (λ0, . . . , λl−1, λl+1, . . . , λd) foreach l = 0, . . . , d. Moreover, we denote by Dαf = ∂|α|f

∂xα11 ∂x

α22 ...∂x

αdd

and by

Dkx−vf =

∑|α|=k

α∈Nd

k!α!

(x − v)αDαf (2.2)

the k-th order directional derivative of f along the line segment between xand v. Finally, we use the notations

Dijf = (vi − vj) · ∇f, i, j = 0, 1, . . . , d (2.3)

for the derivative of f along the directed line segment from xj to xi (as usual,· denotes the dot product) and

Dαl = Dα1

0,lDα21,l . . . Dαl

l−1,lDαl+1l+1,l . . . D

αd

d,l , l = 0, 1, . . . , d, (2.4)

for the composition of derivatives along the directed sides of the simplex.Under these assumptions, we get the following result.

Lemma 2.1. Let f ∈ Cr(S), then

Dkx−v l

f(vl) =∑

|α|=k

α∈Nd

k!α!

Dαl f(vl)λα

l (x), l = 0, 1, . . . , d, (2.5)

for any k ∈ N, k ≤ r and x ∈ Rd.

Proof. Due to the properties satisfied by the barycentric coordinates, for anyl = 0, 1, . . . , d, we have

x − vl = vlλl(x) − vl +d∑

j=0j �=l

vjλj(x)

= vl (λl(x) − 1) +d∑

j=0j �=l

vjλj(x)

= −vl

⎛⎜⎜⎝

d∑j=0j �=l

λj(x)

⎞⎟⎟⎠ +

d∑j=0j �=l

vjλj(x)


=d∑

j=0j �=l

(vj − vl)λj(x)

= (v0 − vl, . . . ,vl−1 − vl,vl+1 − vl, . . . ,vd − vl) · λl(x). (2.6)

By substituting (2.6) in (2.2) and by definition (2.4), we have

Dkx −vl

f(vl) =∑

|α|=k

α∈Nd

k!

α!((v0 − vl, . . . , v l−1 − v l, v l+1 − v l, . . . , vd − vl) · λl(x))

αD

αf(vl)

=∑

|α|=k

α∈Nd

k!

α!((v1 − vl, . . . , vl−1 − v l, v l+1 − v l, . . . , vd − vl) Df(vl))

αλl(x)

α

=∑

|α|=k

α∈Nd

k!

α!D

αl f(v l)

αλl(x)

α.

�

Proposition 2.2. Let S ⊂ Rd be a not degenerated d-dimensional simplex with

vertices v0, . . . ,vd then∫

S

λα(x)dx =A(S)α!

(d + |α|)! , α ∈ Nd+1. (2.7)

Proof. See [15, Theorem 2.2]. �


vertices v0,v1, . . . ,vd. For any x ∈ S and r ∈ N, we have

∫S

d∑i=0

‖x − vi‖r+2λi(x)dx ≤ A(S)(d + 2)!

d∑i=0

d∑j=0

‖vi − vj‖r+2. (2.8)

Proof. By the equality (2.6) and by recalling that 0 ≤ λj(x) ≤ 1 for eachx ∈ S, we easily obtain

‖x − vl‖r+2 ≤⎛⎝ d∑

j=0

‖vj − vl‖λj(x)

⎞⎠

r+2

≤d∑

j=0

‖vj − vl‖r+2λj(x)

for any l = 0, . . . , d. Consequently,

∫S

d∑i=0

‖x − vi‖r+2λi(x)dx ≤∫

S

d∑i=0

d∑j=0

‖vj − vi‖r+2λj(x)λi(x)dx

≤d∑

i=0

d∑j=0

‖vj − vi‖r+2

∫S

λj(x)λi(x)dx

and, by Proposition 2.2, we easily get the inequality (2.8). �


2.2. Construction of the Quadrature Formula

The multivariate Lagrange interpolation polynomial on the simplex S inbarycentric coordinates is

LS [f ](x) =d∑

l=0

λl(x)f(vl). (2.9)

The operator LS reproduces polynomials up to the degree 1 and interpolatesthe values of f at the vertices vl of the simplex S. If the function f belongsto Cr(S), we can replace the values f(vl) by the modified Taylor polynomialof degree r at v� proposed in [6], the resulting polynomial operator is

LSr [f ](x) =

d∑l=0

(r∑

k=0

ark

k!Dk

x−v lf(vl)

)λl(x), x ∈ R

d, (2.10)

where ark = (1+r−k)!r!(1+r)!(r−k)! . As specified in [6], the operator LS

r [f ](x) reproducespolynomials up to the degree max {1, 1 + r} = 1+r. Moreover, for each x ∈ Sand f ∈ Cr+2 (S), its remainder term RS

r [f ](x) = f(x) − LSr [f ](x) can be

explicitly represented as

RSr [f ](x) =

d∑l=0

λt(x)∫ 1

0

−t(1 − t)r

(1 + r)!Dr+2

x−v lf (vl + t (x − vl)) dt. (2.11)

Remark 2.4. Since S is a compact convex domain and LS is a linear boundedoperator, in line with [8], LS

r can be interpreted as

LSr [f ](x) = LS

[r∑

k=0

ark

k!Dk

x−·f

](x)

and, from [8, Proposition 3.4], it follows that LSr 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,

LSr [f ](x) =

d∑l=0

⎛⎜⎜⎝

r∑k=0

ark

k!

∑|α|=k

α∈Nd

k!α!

Dαl f(vl)λα

l (x)

⎞⎟⎟⎠ λl(x)

=d∑

l=0

⎛⎜⎜⎝

r∑k=0

(1 + r − k)!r!(1 + r)!(r − k)!

∑|α|=k

α∈Nd

1α!

Dαl f(vl)λα

l (x)

⎞⎟⎟⎠ λl(x)

=d∑

l=0

⎛⎜⎜⎝

r∑k=0

1 + r − k

1 + r

∑|α|=k

α∈Nd

1α!

Dαl f(vl)λα

l (x)

⎞⎟⎟⎠ λl(x) (2.12)


and, by the change of dummy index, we get

LSr [f ](x) =

r∑k=0

1 + r − k

1 + r

⎛⎜⎜⎝

d∑l=0

∑|α|=k

α∈Nd

1α!

Dαl f(vl)λα

l (x)λl(x)

⎞⎟⎟⎠ . (2.13)

The quadrature formula is then computed by integrating the right hand sideof (2.13) on the simplex S.

Theorem 2.5. Let f ∈ Cr+2(S). Then∫S

f(x)dx = QSr [f ] + ES

r [f ],

where

QSr [f ] =

A(S)(1 + r)

r∑k=0

1 + r − k

(d + 1 + k)!

⎛⎜⎜⎝

d∑l=0

∑|α|=k

α∈Nd

Dαl f(vl)

⎞⎟⎟⎠ (2.14)

and

ESr [f ] =

∫S

RSd [f ](x)

=∫

S

d∑l=0

(∫ 1

0

−t(1 − t)r

(1 + r)!Dr+2

x−v lf (vl + t (x − vl)) dt

)λl(x)dx.

(2.15)

Moreover, the quadrature formula QSr [f ] has degree of exactness 1 + r .

Proof. By integrating the right hand side of equality (2.13), we get

QSr [f ](x) =

∫S

LSd [f ](x)dx

=r∑

k=0

1 + r − k

(1 + r)

⎛⎜⎜⎝

d∑l=0

∑|α|=k

α∈Nd

1α!

Dαl f(vl)

∫S

λαl (x)λl(x)dx

⎞⎟⎟⎠ .

(2.16)

By Proposition (2.2)∫S

λαl (x)λl(x)dx =

A(S)α!(d + 1 + |α|)!

and then (2.16) becomes

QSr [f ] =

r∑k=0

1 + r − k

(1 + r)

⎛⎜⎜⎝

d∑l=0

∑|α|=k

α∈Nd

1α!

Dαl f(vl)

A(S)α!(d + 1 + |α|)!

⎞⎟⎟⎠ . (2.17)


The expression of ESr [f ] is obtained by integrating on the simplex S the

remainder term RSr [f ](x) in formula (2.11). Since RS

r [f ](x) vanishes wheneverf is a polynomial in d variables of total degree at most 1 + r, ES

r [f ] inheritsthis property and the quadrature formula has degree of exactness 1 + r. �2.3. Error Bounds

To give a bound for the remainder term ESr [f ] of the quadrature formula in

Theorem 2.5, we need some additional notations. More precisely, for a k-timescontinuous differentiable function f : S → R, we introduce the norm∣∣Dkf

∣∣S

:= supx∈S

sup{∣∣Dkyf(x)

∣∣ : y ∈ Rd, ‖y‖ = 1}. (2.18)

where ‖ · ‖ denotes the Euclidean norm in Rd, and y is assumed to be a

column vector. Consequently, for any x ∈ S and y ∈ Rd, we have

|Dkyf(x)| ≤ |Dkf |S · ‖y‖r+2. (2.19)


vertices v0, . . . ,vd and f ∈ Cr+2(S). Then

∣∣ESr [f ]

∣∣ ≤∣∣Dr+2f

∣∣S

(r + 2)!(r + 1)A(S)

(d + 2)!

d∑l=0

d∑j=0

‖vl − vj‖r+2. (2.20)

Proof. By taking the modulus of both sides of equality (2.15), by applyingthe triangular inequality and by bounding the directional derivative of f oforder r + 2 by (2.19), we have

∣∣ESr [f ]

∣∣ ≤∣∣∣∣∣∫

S

d∑l=0

(∫ 1

0

−t(1 − t)r

(1 + r)!Dr+2

x−v lf (vl + t (x − vl)) dt

)λl(x)dx

∣∣∣∣∣≤ ∣∣Dr+2f

∣∣S

∫S

d∑l=0

(∫ 1

0

t(1 − t)r

(1 + r)!‖x − vl‖r+2dt

)λl(x)dx

≤∣∣Dr+2f

∣∣S

(1 + r)!

∫S

d∑l=0

‖x − vl‖r+2

(∫ 1

0

t(1 − t)rdt

)λl(x)dx.

Using the inequality in Proposition (2.3), and by the fact that∫ 1

0

t(1 − t)rdt =1

(r + 2)(r + 1),

we have∣∣ES

r [f ]∣∣ ≤

∣∣Dr+2f∣∣S

(1 + r)!(r + 1)(r + 2)A(S)

(d + 2)!

d∑l=0

d∑j=0

‖vl − vj‖r+2

and then (2.20). �Remark 2.7. It is worth noting that Theorem 2.5 gives a quadrature formulaobtained by integrating both sides of the expression in [6, Theorem 1] andthe bound in Proposition 2.6 is nothing but the integral of the bound givenin [6, Theorem 2], where Ω = S, m = 1 and φi (x) = λi(x). Consequently,the bound (2.20) is the best possible estimation for each r ∈ N0.


3. Integration Formulas on the Simplex with Only FunctionData

The main feature of the quadrature formula (2.14) is that it uses only deriva-tives of f along the edges of S; this motivates to consider approximations ofthose derivatives to obtain quadrature formulas which use the function dataat the vertices of the simplex S, at points on its facets or at its center ofgravity. To this aim, we focus on the case r = 1 and we consider differentkinds of approximation of the derivatives in (2.14). For r = 1, the quadratureformula (2.14) in Theorem 2.5 becomes

QS1 [f ] =

A(S)(d + 1)!

d∑l=0

f(vl) +A(S)

2(d + 2)!

d∑l=0

∑|α|=1

α∈Nd

Dαl f(vl). (3.1)

Proposition 3.1. Let f : S → R be a 3-times continuous differentiable func-tion on S, then

QS1 [f ] =

A(S)(d + 2)!

((3d + 2)

d∑l=0

f(vl) − 4d−1∑l=0

d∑r=l+1

f

(vr + vl

2

))

+A(S)

2(d + 2)!

d−1∑l=0

d∑r=l+1

εl,r. (3.2)

where

εl,r[f ] = D3vl−vr

f(vr + ξ1(vl − vr))

−D3vl−vr

f(vr + ξ2(vl − vr)), ξ1, ξ2 ∈ [0, 1].

Moreover, the quadrature formula (3.2) has degree of exactness 2.

Proof. By definition (2.4), the sum of first-order derivatives in the secondterm of QS

1 [f ] can be rewritten as

d∑l=0

∑|α|=1

α∈Nd

Dαl f(vl) =

d∑l=0

d∑r=0r �=l

Drlf(vl) =d−1∑l=0

d∑r=l+1

(Dlrf(vr) − Dlrf(vl)) ,

(3.3)

where the differences of directional derivatives along the edges of the simplexS can be replaced by a three-point finite difference approximation. To dothis, let us recall that for a univariate function g, it is possible to considerthe derivation formula

g′(a − h) =1h

(−1

2g(a + h) + 2g(a) − 3

2g(a − h)

)+

h2

3g′′′(ξ) (3.4)

for some ξ ∈ [a − h, a + h]. Using this formula with h = ±1/2 and a = 1/2we 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, Section5.1]. By applying the formula (3.4) along the edges of S we get

Dlrf(vr) − Dlrf(vl) = 4(

f(vr) − 2f

(vr + vl

2

)+ f(vl)

)+ εl,r (3.5)

with

|εl,r| ≤ 112

ω(D3

l f ((1 − t)vl + tvr) , 1),

where ω denotes the modulus of continuity with respect to t ∈ [0, 1]. Bysubstituting expression (3.5) in (3.3) and by rearranging, we get

d∑l=0

∑|α|=1

α∈Nd

Dαl f(vl) = 4d

d∑l=0

f(vl) − 8d−1∑l=0

d∑r=l+1

f

(vr + vl

2

)+

d−1∑l=0

d∑r=l+1

εl,r.

(3.6)

Finally, by substituting (3.6) in (3.1), we get

QS1 [f ] =

A(S)(d + 2)!

((3d + 2)

d∑l=0

f(vl) − 4d−1∑l=0

d∑r=l+1

f

(vr + vl

2

))

+A(S)

2(d + 2)!

d−1∑l=0

d∑r=l+1

εl,r.

εl,r[f ] = 0 whenever f is a polynomial in d variables of total degree 2 andthis implies that QS

1 [f ] has degree of exactness 2. �


vertices (vl)l=0,1,...,d. Let us denote by sd−1l , l = 0, 1, . . . , d the facet opposite

to the vertex vl and by gl the barycenter of sd−1l . For all α ∈ (0, 1) we have

∫S

f(x)dx =A(S)

2(d + 2)!

(α(d + 4) − d

α

d∑l=0

f(vl) +d

α − α2

d∑l=0

f(yl(α))

+αd

α − 1

d∑l=0

f(gl)

)+ R(d, α)[f ],

(3.7)

with yl(α) = vl + α(gl − vl) and

R(d, α)[f ] =dA(S)

4(d + 2)!(α − α2)

d∑l=0

(α2

∫ 1

0

(1 − t)2D3g l−v l

f(v l + t(g l − v l))dt

−∫ α

0

(α − t)2D3g l−v l

f(v l + t(g l − v l))dt

). (3.8)

The quadrature formula (3.7) has degree of exactness 2.


Proof. Let gl be the barycenter of sd−1l , by Lemma (2.1) the sum of first-order

derivatives along the edges of S in (3.1) can be rewritten asd∑

l=0

∑|α|=1

α∈Nd

Dαl f(vl)λα(gl) =

d∑l=0

Dg l−v lf(vl)

and, since λk(gl) = 1d , for each l, k = 0, . . . , d, then

d∑l=0

∑|α|=1

α∈Nd

Dαl f(vl) = d

d∑l=0

Dg l−v lf(vl). (3.9)

By substituting (3.9) in (3.1), we get

QS1 [f ] =

A(S)(d + 1)!

d∑l=0

f(vl) +dA(S)

2(d + 2)!

d∑l=0

Dg l−v lf(vl). (3.10)

To have a three point finite difference approximation of the directional deriva-tives in (3.10), for each l = 0, . . . , d, let us introduce the univariate function

hl :{

[0, 1] → R,t �→ hl(t) = f(vl + t(gl − vl)).

(3.11)

For t = 1 and t = α ∈ (0, 1), the second-order Taylor expansion of hl(1) andhl(α) centered at 0 with integral remainder are

hl(1) = hl(0) + h′l(0) +

12h

′′l (0) +

12

∫ 1

0

(1 − t)2h′′′l (t)dt (3.12)

and

hl(α) = hl(0) + αh′l(0) +

12α2h

′′l (0) +

12

∫ α

0

(α − t)2h′′′l (t)dt. (3.13)

Then, by (3.12) and (3.13), we get

hl(α) − α2hl(1) =(1 − α2

)hl(0) + (α − α2)h′

l(0) + Rl[f ](α) (3.14)

where

Rl[f ](α) =12

∫ α

0

(α − t)2h′′′l (t)dt − α2 1

2

∫ 1

0

(1 − t)2h′′′l (t)dt.

Therefore, by (3.14) it follows that

h′l(0) =

1α − α2

hl(α) − α

1 − αhl(1) − 1 + α

αhl(0) +

1α − α2

Rl(α)[f ].

(3.15)

By rewriting equality (3.15) in terms of f we get

Dg l−v lf(vl) =

1α − α2

f (αgl + (1 − α)vl) − α

1 − αf(gl) − 1 + α

αf(vl)

+1

α − α2Rl(α)[f ], (3.16)


where

Rl(α)[f ] =12

(α2

∫ 1

0

(1 − t)2D3g l−v l

f(vl + t(gl − vl))dt

−∫ α

0

(α − t)2D3g l−v l


). (3.17)

Finally, by substituting (3.16) in (3.1), we get (3.7). R(α)[f ] = 0 whenever fis a polynomial in d variables of total degree 2 and, therefore, the quadratureformula (3.7) has degree of exactness 2. �

Remark 3.3. 1. For α =d

d + 4the formula (3.7) becomes

∫S

f(x)dx =A(S)

2(d + 2)!

((d + 4)2

4

d∑l=0

f

(dgl + 4vl

d + 4

)− d2

4

d∑l=0

f(gl)

)

+R

(d,

d

d + 4

)[f ], (3.18)

that is a quadrature formula which uses only the function data at thepoints gl and dg l+4v l

d+4 , l = 0, . . . , d, and is exact for all polynomial ofdegree less than or equal to 2.

2. For α = dd+1 , the quadrature formula (3.7) becomes,

∫S

f(x)dx =A(S)

2(d + 2)!

(3

d∑l=0

f (vl) + (d + 1)3f(x∗) − d2d∑

l=0

f(gl)

)

+R

(d,

d

d + 1

)[f ] (3.19)

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 Simp-son rule for a simplex proposed in [7, Theorem 5.1]. For a particular valueof the parameter of the linear combination, we are able to get a quadratureformula with an higher degree of exactness.

Corollary 3.4. Let f : S → R be a 3-times continuous differentiable functionon S. Let us denote by x∗ the center of gravity of S, by sd−1

l , l = 0, 1, . . . , d

the facets of S opposite to the vertex vl and by gl the barycenter of sd−1l .

Then, ∫S

f(x)dx = τF1[f ] + (1 − τ)F2[f ] + R(τ)[f ], τ ∈ R, (3.20)

where

F1[f ] =A(S)

d!

(d + 1d + 2

f(x∗) +1

(d + 1)(d + 2)

d∑l=0

f(vl)

)


is the multivariate Simpson rule for a simplex [7, Theorem 5.1],

F2[f ] =A(S)

2(d + 2)!

(3

d∑l=0

f (vl) + (d + 1)3f(x∗) − d2d∑

l=0

f(gl)

)

is given by (3.7) for α = dd+1 and

R(τ)[f ] = τRSid [f ] + (1 − τ)R

(d,

d

d + 1

)[f ] (3.21)

with RSid [f ] denoting the remainder term in the multivariate Simpson rule for

a simplex. For all τ ∈ R we have R(τ)[f ] = 0, whenever f is a polynomial ind variables of total degree at most 2.

Proof. Since RSid [f ] = 0 and R

(d, d

d+1

)[f ] = 0 whenever f is a polynomial

in d variables of total degree less than or equal to 2, it easily follows thatR(τ)[f ] vanishes for each polynomial of degree at most 2. �

For τ = 3(d+1)d+3 , the family of quadrature formulas (3.20) yields to a

formula which has degree of exactness 3.

Theorem 3.5. Let f : S → R be a 3-times continuous differentiable functionon S and let us denote by x∗ the center of gravity of S, by sd−1

l , l = 0, 1, . . . , d

the facet opposite to the vertex vl and by gl the barycenter of sd−1l . The

quadrature formula∫

S

f(x)dx =A(S)

(d + 3)!

(3

d∑l=0

f(vl) + d3d∑

l=0

f(gl) + (d + 1)3(3 − d)f(x∗)

)

+R

(3(d + 1)d + 3

)[f ], (3.22)

with R(

3(d+1)d+3

)[f ] defined in (3.21), has degree of exactness 3.

Proof. For τ = 3(d+1)d+3 , the quadrature formula (3.20) reduces to

∫S

f(x)dx ≈ F3[f ]

=A(S)

(d + 3)!

(3

d∑l=0

f(vl) + d3d∑

l=0

f(gl) + (d + 1)3(3 − d)f(x∗)

)

(3.23)

and, by Corollary 3.4, it follows that F3[f ] has degree of exactness 2. LetP3(x) be a polynomial in d variables of degree 3; we can write P3(x) as

P3(x) = P2(x) +d∑

i=0

cix3i +

d∑i=0

d∑j=0j �=i

bijx2i xj +

d∑i=0

d∑j=0j �=i

d∑k=0

k �=i,j

dijkxixjxk


where P2(x) is a polynomial of degree 2. Therefore, it is sufficient to provethat F3[·] is exact for the monomials

M1 = x3i ; M2 = x2

i xj (j = i); M3 = xixjxk (k = i, j) i, j, k = 0, . . . , d.

Thanks to the linear isomorphism which maps the standard simplex Δd ofR

d to a generic simplex S, without loss of generality, we can restrict to thecase of the simplex Δd of vertices v0 = (0, . . . , 0); v1 = (1, 0, . . . , 0); vl =(0, 0, . . . , 1, . . . , 0); vd = (0, . . . , 0, 1). The center of gravity of Δd is x∗ =(

1d+1 , . . . , 1

d+1

)and the barycenter of the facets are

g0 =(

1d , 1

d , . . . , 1d

), g1 =

(0, 1

d , . . . , 1d

), . . . , gd =

(1d , 1

d , . . . , 0).

Let us now consider M1 = x3i , i = 0, . . . , d. The exact integral of M1 over the

simplex Δd is [11] ∫Δd

M1dx =6

(d + 3)!,

and, in addition,d∑

l=0

f(vl) = 1,

d∑l=0

f(gl) =d

d3,

f(x∗) = 1(d+1)3 .

(3.24)

By substituting equalities (3.24) in the quadrature formula (3.23) we have

F3[M1] =A(Δd)(d + 3)!

(3 + d3 d

d3+ (d + 1)3(3 − d)

1(d + 1)3

)

=6

(d + 3)!=

∫Δd

M1dx.

Let us consider M2 = x3i xj , i, j = 0, . . . , d and j = i. The exact integral of

M2 over the simplex Δd is [11]∫Δd

M2dx =2

(d + 3)!,

and, in addition,d∑

l=0

f(vl) = 0,

d∑l=0

f(gl) =d − 1d3

,

f(x∗) = 1(d+1)3 .

(3.25)

By substituting equalities (3.25) in the quadrature formula (3.23) we have

F3[M2] =A(Δd)(d + 3)!

(0 + d3 d − 1

d3+ (d + 1)3(3 − d)

1(d + 1)3

)


=2

(d + 3)!=

∫Δd

M2dx.

Finally, let us consider M3 = xixjxk, i, j, k = 0, . . . , d and k = i, j for whichthe exact integral over the simplex Δd is [11]∫

Δd

M3dx =1

(d + 3)!

andd∑

l=0

f(vl) = 0,

d∑l=0

f(gl) =d − 2d3

,

f(x∗) = 1(d+1)3 .

(3.26)

By substituting equalities (3.26) in the quadrature formula (3.23), we have

F3[M3] =A(S)

(d + 3)!

(0 + d3 d − 2

d3+ (d + 1)3(3 − d)

1(d + 1)3

)

=1

(d + 3)!=

∫Δd

M3dx.

Then ∫Δd

P3(x)dx = F3[P3]

and this shows that the degree of exactness of the quadrature formula (3.22)is 3. �

4. The 2-Simplex Case

In this section, we restrict to the case d = 2 in which S is a triangle of verticesv0, v1, v2. In this particular case, the quadrature formula (2.14) reduces to

QSr [f ] =

1(1 + r)

r∑k=0

1 + r − k

(3 + k)!

k∑j=0

(Dj

v1−v0Dk−j

v2−v0f(v0)

+Djv0−v1

Dk−jv2−v1

f(v1)

+Djv0−v2

Dk−jv1−v2

f(v2))

(4.1)

and, by easy computations, the bound for the approximation error becomes∣∣EΔ2

r [f ]∣∣ ≤

∣∣Dr+2f∣∣S

12(r + 2)!(1 + r)

(1 +

√2r

). (4.2)

The quadrature formula (3.19), which has degree of exactness 2 and usesonly function data at the vertices of S, at the midpoints of its sides and atits center of gravity, becomes∫

S

f(x)dx = QS1 [f ] + R

(2,

23

)[f ]


where

QS2 [f ] =

A(S)16

(2∑

l=0

f(vl) − 43

1∑l=0

2∑r=l+1

f (gl) + 9f(x∗)

)(4.3)

and

R

(2,

23

)[f ] =

3A(S)32

2∑l=0

(49

∫ 1

0

(1 − t)2D3g l−v l


−∫ 2

3

0

(23

− t

)2

D3g l−v l


).

The quadrature formula (3.22), which has degree of exactness 3 and usesonly function data at the vertices of S, at the midpoints of its sides and atits center of gravity, becomes

QS3 [f ] =

A(S)40

(d∑

l=0

f(vl) +83

d∑l=0

f(gl) + 9f(x∗)

), (4.4)

where gl =vl + vr

2, i.e. gl is nothing but the midpoint of the side of the

triangle S opposite to vl. For d = 2 and α = 1/3 the formula (3.7), which hasdegree of exactness 2 and uses function data at the midpoints of the sides ofS and at the points g l+2v l

3 , l = 0, . . . , d, becomes∫S

f(x)dx = QS2 [f ] + R

(2,

13

)[f ] (4.5)

with

QS2 [f ] =

A(S)48

(9

2∑l=0

f

(gl + 2vl

3

)−

2∑l=0

f(gl)

)(4.6)

and R(2, 1

3

)[f ] defined in (3.8).

To enhance the degree of exactness of the quadrature formula (4.6), let usconsider the midpoint formula for the 2-dimensional simplex [5]

∫S

f(x)dx =A(S)

6

2∑r=0

f(gr) + E[f ]

where E[f ] = 0 whenever f is a polynomial in 2 variables of total degree atmost 2. We set

Qmid2 [f ] =

A(S)6

2∑l=0

f(gl)

and define the convex combination∫S

f(x)dx = αQS2 [f ] + (1 − α)Qmid

2 [f ] + Eα[f ], α ∈ R. (4.7)

where

Eα[f ] = αR

(2,

13

)[f ] + (1 − α)E[f ]. (4.8)


Since, for all α ∈ R, Eα[f ] = 0 whenever f is a polynomial in 2 variables oftotal degree at least 2, the quadrature formula (4.7) has at least degree ofexactness 2.

Theorem 4.1. Let f : S ⊂ R2 → R be a 3-times continuous differentiable

function on S. Then, the quadrature formula∫S

f(x)dx = MQS3 [f ] + E 4

5[f ] (4.9)

with

MQS3 [f ] =

A(S)20

(2∑

l=0

f

(gl + 2vl

3

)+

13

3∑l=1

f(gl)

)

(4.10)

has degree of exactness 3.

Proof. Equality (4.10) follows by setting α = 45 in (4.7) and by rearranging.

To prove the degree of exactness of the formula (4.9), it is sufficient to followthe same arguments used in the proof of Theorem 3.5 for d = 2. �

Finally, to obtain a quadrature formula over S with degree of exactness4, we consider the convex combination of the quadrature formulas (4.4) and(4.10) ∫

S

f(x)dx = αQS3 [f ] + (1 − α)MQS

3 [f ] + E′α[f ], α ∈ R, (4.11)

where

E′α[f ] = αR

(95

)[f ] + (1 − α)E 4

5[f ],

with R(

95

)[f ] given by the Eq. (3.21) and E 4

5[f ] by the equation (4.8). Since,

for all α ∈ R , E′α[f ] = 0 whenever f is a polynomial in 2 variables of

total degree at least 3, the quadrature formula (4.11) has at least degree ofexactness 3.

Theorem 4.2. Let f : S ⊂ R2 → R be a 3-times continuous differentiable

function on S and x∗ the center of gravity of S. Then, the quadrature formula∫S

f(x)dx = MQS4 [f ] + E′

13[f ] (4.12)

where

MQS4 [f ] =

A(S)120

(2∑

l=0

f(vl) + 42∑

l=0

f(gl) + 12f(

gl + 2vl

3

)+ 9f(x∗)

)

(4.13)

and

E′13[f ] =

13R

(95

)[f ] +

(1 − 1

3

)E 4

5[f ],

has degree of exactness 4.


Table 1. Absolute value of the remainder terms EΔ2r [fi] =

QΔ2r [fi] − ∫

Δ2fi(x)dx, i = 1, . . . , 4; r = 1, . . . , 10

r∣∣EΔ2

r [f1]∣∣ ∣∣EΔ2

r [f2]∣∣ ∣∣EΔ2

r [f3]∣∣ ∣∣EΔ2

r [f4]∣∣

1 2.67e−3 4.39e−2 3.02e−4 1.91e−42 1.06e−3 1.03e−2 3.54e−5 3.87e−53 7.03e−5 1.86e−2 2.55e−6 1.68e−64 1.73e−5 4.35e−3 1.75e−7 1.93e−75 1.18e−6 5.51e−3 1.36e−8 8.93e−96 2.15e−7 4.01e−3 6.76e−10 7.55e−107 1.35e−8 6.67e−4 5.11e−11 3.30e−118 2.60e−9 1.85e−3 2.02e−12 2.28e−129 6.48e−10 3.82e−4 1.43e−13 9.02e−1410 5.07e−10 5.86e−4 5.16e−15 4.94e−15

Table 2. Absolute value of the remainder terms EΔ22 [fi] ,

EΔ23 [fi], EΔ2

MQ3[fi], EΔ2

MQ4[fi] i = 1, . . . , 4

fi

∣∣∣EΔ22 [fi]

∣∣∣∣∣∣EΔ2

3 [fi]∣∣∣

∣∣∣EΔ2MQ3

[fi]∣∣∣

∣∣∣EΔ2MQ4

[fi]∣∣∣

f1 1.55e−3 2.86e−4 1.46e−4 1.12e−6f2 3.85e−3 8.64e−3 4.59e−3 1.83e−4f3 3.71e−4 1.01e−5 5.17e−6 7.89e−8f4 3.76e−4 1.08e−5 5.51e−6 7.91e−8

Proof. Equality (4.13) follows by setting α = 13 in (4.11). To prove the degree

of exactness of the formula (4.12), we proceed by verifying the exactness ofthe quadrature formula for the monomials x4, x3y, xy3, x2y2, y4, similarlyto the proof of Theorem 3.5 for d = 2. �

5. Numerical Results in d = 2

To test the approximation accuracies of the proposed formulas, we considerthe case d = 2 and the standard triangle S = Δ2 of vertices v0 = (0, 0), v1 =(1, 0), v2 = (0, 1). The numerical experiments are conducted by consideringthe following set of test functions [1]

f1(x, y) = cos(√

1 + x2 + y2)

,

f2(x, y) = exp(

− (3x − 2)2 + (3y − 2)2

4

),

f3(x, y) = sin(π

4x +

π

6y)

,

f4(x, y) = sinh(π

4x +

π

6y)

.


In all the experiments, the exact value of the integrals for functions f1 andf2 are computed by assuming as exact the numerical integration performedby Mathematica. In Table 1, we report the absolute value of the remainderterms EΔ2

r [fi] = QΔ2r [fi] − ∫

Δ2fi(x)dx, i = 1, . . . , 4, r = 1, . . . , 10, and, in

Table 2, we display the absolute value of the remainder terms

EΔ22 [fi] = QΔ2

2 [fi] −∫

Δ2

fi(x)dx, i = 1, . . . , 4,

EΔ23 [fi] = QΔ2

3 [fi] −∫

Δ2

fi(x)dx, i = 1, . . . , 4,

EΔ2MQ3

[fi] = MQΔ23 [fi] −

∫Δ2

fi(x)dx, i = 1, . . . , 4,

EΔ2MQ4

[fi] = MQΔ24 [fi] −

∫Δ2

fi(x)dx, i = 1, . . . , 4.

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Filomena Di TommasoDepartment of Mathematics and Computer ScienceUniversity of CalabriaVia P. Bucci, cubo 30A87036 RendeItalye-mail: [email protected]

Benaissa ZerroudiUniversity Ibn TofailKenitraMoroccoe-mail: [email protected]

Received: September 25, 2019.

Revised: March 11, 2020.

Accepted: August 2, 2020.

on some numerical integration formulas on the d ......d α l = d 1 0,l d 2 1...d l − d αl+1...

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