Interpolation to boundary data on the simplex

Computer Aided Geometric North-Holland

43

Design 2 (1985) 43-52

Interpolation simplex

to boundary

data on the

John A. GREGORY

Presented

at Oberwolfach

12 November

1984

Abstract. An explicit representation of a finite dimensional Hermite interpolation polynomial for the simplex in R” is deacrlbed. The basis functions of this scheme are then used in the construction of a C’ transfinite blending function interpolant for the simplex. Keywords. Multivariate

Interpolation.

simplex

interpolation,

blending

function,

transfinite

interpolation

1. Introduction

The main purpose of this paper is to define a transfinite blending function interpolant for the simplex in R”. The term ‘transfinite’ means that the interpolant matches an infinite set of data, namely function and derivative values given on all faces of the simplex. Here the term ‘blending function’ means that the interpolant is constructed as a convex combination (i.e. ‘blend’) of simpler constituents, each of which matches function and derivative values on all but one of the faces of the simplex. Blending function interpolation on the simplex and, in particular, the tetrahedron is considered in [Mansfield ‘761. where compositions of rational Hermite interpolation projectors are defined. In this present paper we propose a scheme which is a composition of polynomial Taylor interpolants. This scheme is a generalization of an interpolant for the triangle described in [Gregory ‘781 and is based on the internal report [Gregory ‘791. One advantage of the scheme is that it has a relatively simple explicit construction. Before describing the transfinite scheme we develop, in Section 3, an explicit representation of a finite dimensional Hermite polynomial interpolant for the simplex. This interpolant matches function and certain derivative values given at the vertices of the simplex and its existence is suggested by the work of [Mansfield ‘761. However, the piecewise application of the interpolant over a tessellation of a domain in R”, only gives rise to a Co function, even though derivatives across the vertices of the tessellation are continuous. The importance of the Hermite interpolant to this paper is that its basis functions can be used to construct polynomial weight functions.for the blending function scheme. Alternatively. rational weight functions can be employed. The blending function scheme is described in general terms in Section 4 and the Taylor interpolation form is developed in Section 5. For an appropriate N, the blending function scheme gives a C”’ function when it is piecewise applied over a tessellation. The interpolation schemes described in this paper define bounded idempotent linear operators, i.e. projectors, on some appropriate function space. Thus the schemes are able to reproduce all functions in the range of the interpolation projector. The range is thus called the precision set of the interpolant. The subset of all polynomials of a certain degree which can be 0167-X396/85/$3.30

0: 1985, Elsevier Science Publishers

B.V. (North-Holland)

44

J.A. Gregory / Intrrpolution

on the simplex

contained in the precision set is important in determining the accuracy of an interpolant and is thus considered in the paper. The paper begins with a summary of notation in Section 2. In particular, the barycentric coordinate system for an n-simplex is introduced, since each interpolant will be described in terms of this invariant system.

2. Preliminary

notation

Let n+l

c

X”

n+

c

h,V,IO
J=l

1

x,=1

(2.1)

/=I

where x=(x1

,...)

define a non-degenerate V,=(U,

simplex

,,...,

The representation

X,,)ER”, in R” with vertices

uJER”,

j=1,2

,...,

n+l.

(2.3)

of x E R” as

nil x=

(2.2)

,I + 1

c

x,y>

where c

/=I

A, = 1,

(2.4)

/=I

uniquely defines the barycentric coordinate system X,=X,(x), j= 1, 2 ,..., n-t 1. Let E, denote a point on the face A, = 0, i.e. the face opposite the vertex I/;. Then E, can be represented as

c

E,=

X,V,, whereh,=O,

c

X,=1.

(2.5)

/=I

J=l

The barycentric coordinate system can be interpreted as defining an affine transformation takes S,,, with vertices V,, between (x1,. . , x,,) E R” and (A,, . , A,,) E R”. This transformation j=l,2 simplex s,,. with vertices 5 = e,, j = 1, 2,. . , n, and ‘..., n + 1, onto a standard C+, = 0 respectively, where e,, j = 1,. , n, denotes the canonical basis of R”. We define a derivative operator along the edge joining the vertices V, and V, by

D,,=

c

(ox,-uk,)a/ax,,

i+j,

,...,n+l,

i,j=1,2

(2.6)

k=l

and a product

of such operators

along all edges which meet at V, by

n+l

D,= n D,,, i=l.2

,...,

n+l.

(2.7)

j=l If'

Furthermore,

if N,, (Ye.. . , a,+, denote (Y,=((Y ,,...,

a,_,,a,+1,...1

n + 1 non-negative a,+1 )?

i=l,2

,...)

integers n+l,

and

(2.8)

45

denotes

a multi-index

of n of these integers,

,I + 1 D,a’ = n D,‘71. i=1.2

then we define

. . . . . n+l.

(2.9)

/=1 J+’

Finally,

with ci= (CI,,...‘(Y,,).

(2.10)

D” = fi

(2.11)

we define (a/ax,)“‘.

We conclude this section with a simple lemma which will be useful in subsequent work. The lemma follows immediately on differentiating (2.12) along the line segment x(8) = (1 - e)y + /Iv,. Lemma 2.1. Lef f(x)=g(X1.

(2.12)

X*.....h,,+,),

Kjhere f is u real differentiable (2.4). Then

of n rxzriub1e.r and A,, j = 1, 2,. . , n + 1, ure defined h?

function

D,,f=(a/ax,-a/ax,)g, i+j, i, j= 1, 2.....n

+ I.

(2.13)

3. A Hermite interpolant for the simplex Theorem 3.1. (The Hermite interpolant). function defined on S,, which is such that )(Y),

i=l,2

(0, l,...,

N}.

(D?f

,...,

Given the non-negative

integer N, let f be u real-valued

ci, E N”,

n+l,

(3.1)

are defined, where N=

Then there exists an interpolation

(3.2) pobnomiul

p, explicitly

defined by

n+l

p(x)

= c KY+’ c P,,W( I=1 a, E N”

(3.3)

DP’(f/h’Y+‘))(U

II+ 1 p,,(X)

=

n /=1 I+,

X;.ya,!,

x=

(A,,

x,

,...,

(3.4)

A,,,,).

which is such thut (DTp)(V,)=(Dylf)(V,),

j=l,2

,...,

n+l,

(Y,EN”.

i=1,2

,...,

n+l,

ol,~N”,

Proof. Since ( D~JA‘~+‘)(V,)=O,

i#j,

it follows from (3.3) that (Dp~P)(v,)

= ~D~~(h:-~~~N,,P~,:!D.;if,h:+~))(V.jj)(li/

(3.5)

46

J.A. Gregor), / Interpolation

for all (Y,E N”. Application

of Leibnitz’s

on the .srmple.x

rule then gives that

B

Now. using Lemma

2.1, 1

,I +

n

(a/ax, -

L=l

a/ax,p

k 'J

if o$ = B,,

= p

otherwise.

\O, Hence. substituting

and reconstituting

the summation

= (D,oll( k~+‘J/A‘;+‘))(

using Leibnitz’s

rule gives

v,)

= (D,Y)(v,). which completes

the proof of the theorem.

The interpolant defined by (3.3) is a natural generalization of Hermite two point Taylor interpolation in one variable, where, with n = 1 and x = X,V, + XIV2 E R. X, + h, = 1, we have N p(x) = A:+’ 1 { A’,/;!}( D;,( f/kl’+‘))( V,) ,=o ,Y

KY;”

c { A’,/!}( r=O

see [Davis ‘63, p.371. Equation

D;,(f/k;+‘))(V,),

(3.6) can be expressed

~i~~=,~~~,(~~)(D;,()(v,)+

(3.6) in the cardinal

;h,h@W’)(r:)~

r=O

basis form (3.7)

where h,(h)

= (1 -,),v+’

g

x”(N+k-i)!/N!i!(k-i)!.

(3.8)

k=,

When n = 2 and N = 1 in (3.3) the tricubic interpolant on a triangle of [Birkhoff ‘711 is obtained. The piecewise application of the interpolant (3.3) over a tessellation in R” gives a Co function, except in the case n = 1 when it is C”. The blending function interpolant of the following sections will, however, give C,” functions for any n. It follows from the theory of finite-dimensional interpolation. and the explicit representation given by (3.3) and (3.4) that the linear functionals defined by (3.1) are linearly independent over the (n + l)(N + I)“-dimensional polynomial space defined by

(3.9)

47

Also, the linear operator

P defined

P [ f ]( x ) = p ( x ) .

by (3.10)

f E c ” ” ( s,, ) ,

where p is given by (3.3). is a projection

of C”~“(S,,) into .F. Thus

P2[fl = P[fl

(3.11)

P[f‘]

(3.12)

and

The following

=f

theorem

for all fE.7. gives more insight

into the nature

of 7.

Theorem

3 .2 . .7 is the space of po~vnomiuls M,hose restriction alotyq an edge joining vertices y and V,, i # j, is u po!ynomial oj”degree 2 N + 1, i.e. if II+ I i 1, ifj Y= \ Q, % lO
un_v taco

(3.13)

then ,F= 9. Proof. Clearly,

f=

from (3.9). YC 2’. Thus we require to prove that 9’~ 7. I, + I where 0 < (Y,+ (Y,< 2N + 1, i f j. n x;l.

i.e. if (3.14)

A=1

then f~.?. The proof of this comprises two parts: (i) Suppose (Y,> N + 1 for an index i. Then, since a, + a/, < 2N + 1, k f i. it follows (Ye< N for k # i, k = 1, 2.. . . n + 1. Now (3.14) can be written as ,I + 1 f=~~+‘~;,-*-’ ,!g e .k#l

that

and since LY,- N - 1 + cyx< N it follows that fey. (ii) Supposecu,~Nforallk=1,2....,n+l.Then~~~~cu,~(n+l)N.Assumefurtherthe inductive hypothesis that f~7 for all (Y~ in (3.14) such that M < Earn (Ye. where 1 < M < (n + l)N. Now consider (Ye such that IL;:: (Ye= M - 1. Then (3.14) can be written as

using either the inductive hypothesis if where IX;:; CY~,= M and (Ye,G N + 1. Thus 1~7 (Ye,< N or part (i) if (Ye,= N + 1. The inductive hypothesis is true for M = (n + 1) N since in this case cyx= N for all k = 1. 2,. . , II + 1 and then (3.14) can be written as f=

[‘;gqj$j

=~~~hV+~?&>

so that f~7. Hence, by induction, the hypothesis This completes the proof of the theorem. The following

corollary

follows immediately

is true for all 0 < M < (n + l)N.

from (3.13).

4x

J.A. Gregop /

Corollary 3.1. The following

Inrerpoirtronon the sinlp1e.y

inclusions hold: (3.15)

(3.16) is the set of polynomials

of degree < K.

It should be noted that Pz,vtz

CT

and YcP~(,~+,,~~.

An ulgebruic identity,. We are now in a position blending function scheme which follows. Let following identity can be derived:

to derive an identity which will be used in the f EF and, using (3.12) the

f(x) = 1. Then

n+l

C a,(x)=1 where ~1,EF

(3.17)

forallxER”.

are polynomial

functions

defined

by t1+ I

a,(x)=X.y+’

c

P,,O)(N-t

1% =

Ia,l)!/N!,

c a,’

(3.18)

/=1

a,rN”

J+’

and the p,, are given by (3.4).

4. A general scheme for blending function interpolation In this section we define a general scheme for interpolating function and derivative values given on all faces of the simplex S,. This blending function scheme is defined in the following theorem. Theorem 4.1 (The general blending function interpolant). real-oalued functions defined on S,, which are such thut (D”p,)(E,)=(D”f)(E,),

j#i,

j=l,2

foralllal=

Let f and p,, i = 1, 2,. . , n + 1, he

,..., i

n+l. (4.1)

a,
k=l

Thus p, interpolutes f and its deriuatives of order N und less on UNfuces of the simplex the face A, = 0. Then

excluding

n+l

p(x)

= C a,(x)p,(x), ,=I

where the u, are given by (3.18) (Dap)(E,)=(Daf)(E,)

(4.2)

xESn9 defines a function p which is such that forallj=l,2

,...,

n+l,

\(Y\
(4.3)

J.A. Gregor)- / Interpolation

Proof. The proof is almost Leibnitz’s rule gives

self-evident,

relying

on the smplrx

49

on (3.17) and (3.18). More formally.

applying

08”,)(E,)(D”-BP,)(E,) c (;$‘c ,=l

=

B
=

c (;j’;g; (DB”,)(E,)(D”-Bf)(E,),

p
I*J

where, since (3.18) contains

the factor

(Dau,)(E,)=O Furthermore,

forall

ky”,

we have used the fact that

\p\ GN.

from (3.17).

‘fl

and hence (4.3) follows. Note. A rational

function

p,(x)

alternative

to the polynomial

function

(3.18) is

= ,,:+‘/cx.;+‘.

(4.4)

Suppose p, = P,[f], f~ H, where P, are bounded linear operators such that P, : H + H, i = 1, 2,. , n + 1, and H is an appropriate space of bounded differentiable functions defined on S,,. Suppose further that

Remarks.

p<[gl(x)

= 0,

i=1,2

)...)

n+l.

(4.5)

for all g E H such that (Pg)(E,)=O,

ICI
j=1,2

,...,

n+l.

(4.6)

Then it follows that P,(Z-P,)[f](x)-0 where I is the identity

forall/GH,

operator,

P( I - P)[f](x) Thus P,, i = 1, 2,. on H. Also if P,[f] i.e. if q

= 0

and, moreover, for all f~

, n + 1, and P defined =f

(4.7) that

H.

bounded

(4.8) idempotent

linear operators,

for all fEq,

is the precision

i.e. projectors,

(4.9)

set of the operator

P,, then, using (3.17), it follows that

n+l

P[f]=f i.e. the precision

forallfE set of P contains

(4.10)

n x, !=I the intersection

of the X,.

50

J.A. Gregory / Inrerpolation on the .rimplex

5. Polynomial

blending function interpolation

scheme

We now consider the blending function scheme defined by Theorem 4.1, where p, = P,[f] and the P, are defined by Boolean sums of polynomial Taylor interpolation projectors. Let f-l+1

I,+ 1

C

E:=

X,V,+(X,+X,)K,

i#j,

k+f.

C h,=l,

(5.1)

k=l

k=l

/

be the point of intersection of the face X, = 0 with the line through x E S,, which is parallel the edge joining V, and V,. Also let C,“‘(S,,), i = 1, 2,. . , n + 1, be the function spaces C,““(S,,)={f]D~~f~C”(S,,)fora11q~N”}. Then Taylor

interpolation

T,‘[flb)

projectors

(5.2)

q’ can be defined

on C,“‘(S,,) by

($/k!)( qf)( E/q. j + i,

= ;

to

j=1,2

n+

,....

1.

(5.3)

x=0

Some properties

of these projectors

are given in the following

lemma:

Lemma 5.1. The Tq,lor projectors defined by (5.3) have the interpolution (DAT,‘[f])(E,)=

i+j,

(L$f)(E,),

k=O,

I....,

N,

properties fEC,““(S,,)

that (5.4)

and the precision set property that q’[f]=f

forullfE_ri4’,

(5:5)

where n+l

q’=

ti

A;

!

n

g,(h,)lg,EC,N”(S,,).O~k~N

/+r. / - a/ah,)g(

E,‘) = 0 for any differentiable

E (P+!)( I)@,) = [(a/ah, - ajax,)’ k’=O =((D,:f)(E;))(E,),

k=O,

l....,

Now when x = E, we have E: = E, and hence the interpolation I?+1

f(x)=X’

,cl

(5.6)

:

Proof. Since A, = 0 at E, and since (a/ax, it follows by use of Lemma 2.1 that (Q:T,‘[f

.

I=1

g,(X,),

OGk’GN,

/#I. J then it follows from (5.1) and Lemma

2.1 that

(~:f >(E;)= ((am,- am)“f j( q) n+l k’! n g,(X,),

=I

if k = k’,

I=1 IZi, J

0.

otherwise.

D;,?)(

function

g,

j(

E;) E,)

N. properties

(5.4) follow. Also, if

51

J.A. Gregoq / Inrerpolutron on the wuple.x

Thus substitution into (5.3) gives the desired precision set result (5.5), which completes the proof of the lemma. The projectors T,‘, j + i, j = 1. 2,. , n + 1, are commutative over C,“‘(S,,). We then have the following theorem: Theorem 5.1 (Boolean

sum interpolants).

The n-fold Boolean sum

?I+1 P, = @ r,‘,

(5.7)

/=1 J+’

tithere T/i @ T,;-T,; t T,; - T’T’ /I /:’

(5.8)

defines u projector on C,“‘?S,,) which is such thut forullk=O.l,....

(D,:P,[f])(E,)=(D,:f)(E,)

undj#i,

N

j=1,2

(5.9)

. . . . . n+l.

Furthermore II

P,[f]=f

forullfE3Cq=

+

1

(5.10)

U 3’. /=I /#I

A proof of this theorem is easily supplied using the Boolean is a consequence of the remainder operator representation

sum theory of [Gordon

‘711 and

,I+ 1 n (Z-T,!).

I-P,=

(5.11)

/=I I*1

where the order of the product operator can be permuted since the T,’ commute. Equations (5.9) imply that P,[ f ] interpolates f and its derivatives of order N and less on all faces of the simplex excluding the face X, = 0. Thus, using Theorem 4.1, we conclude the paper with the following interpolation scheme. Theorem 5.2 (The polynomial

blending

function

interpolant).

n+l P[f

1(x>= c Q,(~,P,[fl(,~), x E &, r=l

n+l

f E n

C,“‘l&J~

defines a blending function interpolant on S,,, where the u, ure defined by (3.18). L?,I(,y+,J_, c%, i = 1, 2,. . ., n + 1. then (4.10) implies thut P[fl

=f

forfll~fE$P,I(,v+I,--,.

(5.12)

I=1

Moreover,

since (5.13)

We refer to (5.12) as a polynomial blending function interpolant since the P, and P involve polynomial weights. In practice the rational weights (4.4) could also be used. In Theorems 5.1 and 5.2 the derivatives DPgf are ‘compatible’. By this we mean that the derivatives do not depend on the order in which the differentiation is performed. This condition allows the commutativity of the projectors T,‘, j f i, j = 1, 2,. . . . n + 1. If f E C” but f @ C,“‘(S,,), then it should be possible to add rational correction terms to (5.12) although we have not considered how this can be done in the general case. An alternative approach, considered in [Gregory ‘791 uses a recursive scheme based on Theorem 4.1, which generates a rational

52

J.A. Gregory / Interpolorion

on the .vrmple.x

interpolant on a simplex of dimension k from those on simplices of dimension this leads to a much more complex scheme than that given by Theorem 5.2.

k - 1. However,

References Birkhoff, G. (1971) Tricuhic polynomial interpolation. Proc. Nat. Acad. Sci. U.S.A. 68. 1162-1164. Davis, P.J. (1963) Inrerpolatron and Approximution, Blaisdell. New York. Gordon, W.J. (1971) Blending function methods of bivariate and multivariate interpolation and approximation. SIAM J. Numer. Anal. 8, 1588177. Gregory. J.A. (1978). A blending function interpolant for triangles. in: D.G. Handscomb. ed., Mulriucrrrute Approximutron. Academic Press, London. Gregory, J.A. (1979). Interpolation to Boundary Data on Simplices, TR/87 Math. and Stat. Dept.. Brunel University, Uxbridge. England. Mansfield, L. (1976) Interpolation to boundary data in tetrahedra with applications to compatible finite elements. J. Math. Anal. Appl. 56, 137-164.