Directory UMM :Journals:Journal_of_mathematics:OTHER:

(1)

ON THE TWO DIMENSIONAL SPLINE INTERPOLATION OF

HERMITE-TYPE

by Mihaela Puscas

Abstract. In [5] the authors gave the definition of the interpolating cubic spline of Hermite-type in two variables: this is a polynomial of degree three in both variables and interpolates the function values and the values of the following derivates

D

0,1

,

D

1,0

,

D

1,1 at the knots of a given rectangular subdivision. We give two similar constructions but we use only the function values and the values of the derivates

D

0,1

,

D

1,0 of the unknown function at the knots. WE can prove similar estimates, but in order to compute the values of our spline function we need fewer algebric operations .

In what follows we fix the region

Ω

=

[

a

,

b

]

×

[

c

,

d

]

and a subdivision

.

of

d

y

...

y

c

,

b

x

...

x

a

=

₀

<

₁

<

_N

=

₀

<

₁

<

_M

=

Ω

Let

]

y

,

y

[

]

x

,

x

[

_i _i₁ _j _j₁

j ,

i

=

+

×

+

Ω

and

h

_i

=

x

_i₊₁

−

x

_i

,

l

_j

=

y

_j₊₁

−

y

_j

.

If u is a function defined on Ω, then u_i(_,r_j,s) =Dr,su(x_i,y_j), where the differential operator

D

r,s has the obvious meaning.

2. We are to determine the spline function S of Hermite-type of two variables with the following properties:

(i) On the rectangle

Ω

_i_,_j we have

β α

β αβ α

β

α ₍ ₎ ₍ ₎

) , ( ) , (

3

4 0 ,

) , (

,

,j i j i j

i x y A x x y y

S y x

S = =

∑

− −

≤ +=

(ii) At the knots we have

1 ;

1 , 0 , , )

,

( (_,,)

, = = + ≤

s r s

r u y x S

Drs _i _i _ir_js ,

i

=

0

,

1

,...,

N

;

j

=

0

,

1

,...,

M

The following theorem can be verified by an easy calculation.

Theorem 2.1. If A_i(_,2_j,2) =0,then there exists a unique spline function satisfying conditions (i), (ii). The function S is continuous on Ω.

(2)

{

}

{

}

))]}

(

))

(

)(

1

)[(

1

(

)]

(

)(

1

[(

{

)

1

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

1

(

)

,

(

, 1 1 , 1 ) 1 , 0 ( , 1 , 1 , ) 1 , 0 ( , ) 1 , 0 ( 1 , 1 , 1 1 , 1 ) 1 , 0 ( 1 , , 1 , ) 0 , 1 ( 1 , 1 4 ) 0 , 1 ( 1 , 3 1 , 1 2 1 , 1 ) 0 , 1 ( , 1 4 ) 0 , 1 ( , 3 , 1 2 , 1 , j i j i j i j j i j i j i j j i j j i j i j i j j i j i j i i j i i j i j i j i i j i i j i j i j i

u

l

t

u

l

t

v

u

l

u

t

u

l

u

t

v

u

h

t

u

h

t

u

t

u

t

v

u

h

t

u

h

t

u

t

u

t

v

y

x

S

+ + + + + + + + + + + + + + + + + + + +

−

+

−

+

−

+

−

+

Φ

+

Φ

+

Φ

+

Φ

+

Φ

+

Φ

+

Φ

+

Φ

−

=

(2.1.) where

).

t

1

(

t

)

t

(

,

)

t

1

(

t

)

t

(

),

t

2

3

(

t

)

t

(

),

t

2

1

(

)

t

1

(

)

t

(

2 ₂ 2 ₃ 2 ₄ 2

1

=

−

+

Φ

=

−

Φ

=

−

Φ

=

−

Φ

Using this form, in order to compute S(x,y) at the point (x,y) in Ωi,j we need 56 algebric operations: 34 additions and 22 multiplications.

Theorem 2.2. If

u

∈

C

1,1

(

Ω

)

, then

)

u

D

(

l

2

1

)

u

D

(

h

8

3

)

y

,

x

(

S

)

y

,

x

(

u

−

≤

ω

1,0

+

ω

0,1

Further, if

u

∈

C

2,2

(

Ω

)

, then

)

u

D

(

l

3

1

)

u

D

(

h

32

1

)

y

,

x

(

S

)

y

,

x

(

u

−

≤

2

ω

2,0

+

2

ω

0,2

Proof. First we suppose that

u

∈

C

1,1

(

Ω

)

and let (x,y)

∈

Ωi,j. Using the theorem for the interpolating cubic splines of Hermite-type in one variable, we have:

) 1 j ,j p ( ) u D ( h 8 3 ) y , x ( u u h ) t ( u h ) t ( u ) t ( u ) t

( p 1,0

) 0 , 1 ( p , 1 i i 4 ) 0 , 1 ( p , i i 3 p , 1 i 2 p , i

1 +Φ +Φ +Φ − ≤ ω = +

Φ + +

By fixing the first variable and applying the Lagrange theorem in the second variable, for the third term of (2.1.) we obtain:

))]}

,

x

(

u

D

u

(

t

))

,

x

(

u

D

u

)(

t

1

)[(

v

1

(

)]

u

)

,

x

(

u

D

(

t

)

u

)

,

x

(

u

D

)(

t

1

[(

v

{

vl

)

v

1

(

1 i 1 , 0 ) 1 , 0 ( j , 1 i i 1 , 0 ) 1 , 0 ( j , i ) 1 , 0 ( 1 j , 1 i 1 i 1 , 0 ) 1 , 0 ( 1 j , i i 1 , 0 j

η

−

+

η

−

+

−

η

+

−

η

−

+ + + + + +

where η,η∈(y_jy_j₊₁). Hence, using this and the form (2.1.), we have:

) u D ( l 2 1 ) u D ( h 8 3 ) u D ( vl ) v 1 ( )] y , x ( u ) y , x ( u [ v )] y , x ( u ) y , x ( u )[ v 1 ( ) u D ( h 8 3 ) y , x ( S ) y , x ( u 1 , 0 0 , 1 1 , 0 j 1 j j 0 , 1 ω + ω ≤ ω − + + − + − − + ω ≤ − +

In the case

u

∈

C

2,2

(

Ω

)

we fix one variable of the function u and apply the Taylor formula of the second order and the estimation

) u D ( h 1 ) y , x ( u u h ) t ( u h ) t ( u ) t ( u ) t

( _i_,_p ₂ _i ₁_,_p ₃ _i _i(1_,_p,0) ₄ _i _i(1,₁0_,_p) _p 2 2,0

1 +Φ +Φ +Φ − ≤ ω

(3)

By the Taylor formula and the continuity of D0,2, we can express the third term of (2.1.) in the form:

) , ( u D vl ) v 1 ( 2 1 )} ~ , ~ ( u D ) v 1 ( ) , ( u vD { vl ) v 1 ( 2 1

)]} , x ( u tD ) , x ( u D ) t 1 )[( v 1 ( )] ~ ~ , x ( u tD ) ~ , x ( u D ) t 1 [( v { vl ) v 1 ( 2 1

2 , 0 2 j 2

, 0 2

, 0 2 j

1 i 2 , 0 i

2 , 0 1

i 2 , 0 i

2 , 0 2

j

ζ ξ −

− = ζ ξ −

+ ζ ξ −

− =

= η +

η −

− + η +

η −

−

− + +

where

(

ξ

,

ζ

)

∈

Ω

_i_,_j

.

By these considerations we have:

( )

x y S

( )

x y h

( )

D u u i j

0 , 2 2

,

32 1 ,

, − ≤ −

ω

+

( )

−

[

( )

−

( )

]

+

[

(

₊

)

−

( )

]

−

( )

1

−

( )

ξ

,

ζ

≤

2

1

,

1

v

u

x

y

_j

u

x

y

v

u

x

y

_j ₁

u

x

y

v

vl

2_j

D

0,2

u

( )

D

u

( ) ( )

v

l

( )

D

u

h

( )

D

u

l

( )

D

u

h

0,2

2 0

, 2 2 2

, 0 2 0

, 2 2

9

3

32

1

2

1

32

1

ω

− − −

−

+

≤

+

−

+

≤

,

which proves our statement .

3. We are to determine the spline function S of Hermite-type of two variables with the following properties:

(i’) On the rectangle Ωi,j we have:

β ≤

β + αβ= α

α β

α ₋ ₋

=

=S (x,y)

∑

A (x x ) (y y ) )

y , x (

S _j

2

3 0 ,

i ) , (

j , i j

, i

(ii’) At the knots we have

(

i

=

0

,...,

N

;

j

=

0

,...,

M

)

) 1 , 0 (

j , p j p j , i 1 , 0

) 0 , 1 (

q , i q i j , i 0 , 1

q , p q p j , i

u

)

y

,

x

(

S

D

u

)

y

,

x

(

S

D

u

)

y

,

x

(

S

=

where p = i, i+1 ; q = j, j+1.

The following theorem can be verified by an easy calculation .

Theorem 3.1. There exists a unique spline function S with the properties (i’), (ii’) and it is continuous on Ω.

By the substitution

(

)

(

)

j j i

i

l y y v h x x

(4)

[

( )

(

)]}

(

1

)

{

)(

1

)

(

)

}.

(

3

.

1

.)

{

)]}

(

[

){

1

(

)

,

(

, 1 1 , 1 ) 1 , 0 (

, 1 ,

1 , ) 1 , 0 (

, )

0 , 1 (

1 , 1 , 1 , 1

0 , 1 1 , 1 , ) 0 , 1 (

, , , 1 ) 0 , 1 (

, , ,

t

u

l

t

u

l

v

u

h

u

t

u

h

t

u

v

u

h

u

t

u

h

t

u

v

y

x

S

j i j i j i j j

i j i j i j j

i i j i j i

j i i j i j

i i j i j i j i i j i j

i

+ + + + +

+ + + +

+ +

+

−

+

−

+

−

+

−

+

−

+

−

=

In order to compute the value S(x,y) = Si,j (x,y) at the print (x,y)

∈

Ωi,j we need 16 multiplications and 21 additions.

Lemma 3.1. Let f be differentiable on [a,b], and let for x∈[x_i,x_i₊₁].

.

)

x

](

h

)

x

(

'

f

)

x

(

f

)

x

(

f

[

h

1

)

x

)(

x

(

'

f

)

x

(

f

)

x

(

S

₂ _i ₁ _i _i _i _i 2

i i i

i

=

+

−

+

−

If

(

'

)

3

2

)

(

)

(

],

,

[

1

f

h

x

S

x

f

then

b

a

C

f

∈

−

_i

<

ω

If

(

"

)

1000

46

)

(

)

(

],

,

[

2

f

h

x

S

x

f

then

b

a

C

f

∈

−

i

≤

ω

.

Proof. By the substitution

(

)

i

( )

i i

( )

i i

i

f

x

f

x

h

x

t

=

−

,

=

,

'

=

' we have

)

x

(

f

)

t

(

f

h

t

f

)

t

1

(

f

)

x

(

f

)

x

(

S

)

x

(

R

=

_i

−

=

_i

−

2

+

_i₊₁ 2

+

_i _i'

−

2

−

.

For

f

∈

C

1

[

a

,

b

]

the Lagrange theorem for the interval [x_i,x_i₊₁] gives:

)],

(

'

f

)

(

'

f

)[

t

1

)(

t

1

(

t

h

)]

f

t

)

(

'

f

(

)

t

1

)(

(

'

f

)[

t

1

(

t

h

)

t

1

(

t

f

h

t

)

t

1

(

h

)

(

'

f

)

t

1

(

t

h

)

(

'

f

)

x

(

R

1 i

' i 2

1 i

' i i 2 i

2 2

i 1

ξ

−

ξ

+

−

=

+

ξ

+

ξ

−

=

−

+

−

ξ

+

−

ξ

−

=

where ξ₁,ξ₂,ξ∈(x_i,x_i₊₁). Hence we have

) ' ( 3 3

2 ) ' ( 3 3

2 )

' ( ) 1 )( 1 ( ) ( )

(x f x ht t t f h f h f

Si − ≤ i − + ωi ≤ i ωi ≤ ω

For

f

∈

C

2

[ ]

a

,

b

we substitute the values

f

_i

,

f

_i₊₁

,

f

_i' from the Taylor formula at the point x=x_i +th_i and get

( ) ( )( )(

x

1

t

[

1

t

v

x

)

th

]

f

"

( )

v

dv

t

(

x

v

) ( )

f

"

v

dv

R

x _i₁

x 2 i

i x

x

1 i i

−

+

−

+

−

=

∫

+

In both integrals we let τ=

(

v−x_i

)

/h_i. Then

( )

x

h

( ) (

t

f

"

x

h

)

d

( ) (

t

,

f

"

x

h

)

d

,

R

1 _i _i

t 2 i

i t

0 1 1 2

i





Ψ

τ

+

τ

+

Ψ

τ

+

τ





=

∫

where Ψ₁

( ) ( )( )

t,τ = 1−t

[

1+t τ−t

]

,

Ψ

₂

( ) ( )

t

,

τ

=

t

2

1

−

τ

.

(5)

( ) (

)

( ) ( )

( )

( ) ( )

( )

, 1 2

1 " 2 "

1 2

1 " 1

2 1 " ,

" ,

" "

,

4 2

1 0

1 0 1

t t t f t f

t t t f t

t t f d t f

d t f

d h x f t

t

i i t

+− +

=

= +− +

−− −

= Ψ

+ Ψ

= +

Ψ

∫

∗

η

ξ

η

ξ

τ

η

τ

ξ

τ

τ τ

where

ξ

,

η

∈

[

x

_i

,

x

_i₊₁

]

.

The function Ψ₂

( )

t,τ does not change sign for τ∈

[ ]

t,1 ; hence

( ) (

)

( ) ( )

1

2

"

,

2 2 1

2

t

f

d

h

x

f

t

_i _i

t

−

=

+

Ψ

∫

τ

ζ

where

ζ

∈

[

x

_i

,

x

_i₊₁

]

.

By these

considerations we have:

( )

( ) ( ) ( )

( )

[

"

( )

"

( )

]

,

1 2 1 1

" 1

" 2

1 2 2

2 4

2

ς

ξ

η

f f

t t t h t t f t t f t t f t h x

R _i _i −

+ − =    

 

+ −

  



+ +

− −

=

where

η

∈

[

x

_i

,

x

_i₊₁

]

,

and we have used that

( )

,

and

f

"

t

1

t

sgn

t

1

t

sgn

4 2

+

=

−

is

continuous.Finally,

( )

"

1000

46

"

046

,

0

"

4

11

5

2

f

h

f

h

f

h

x

R

≤

_i

−

η

−

η

≤

_i

ω

_i

≤

ω

and lemma is

proved.

Theorem 3.2. If

u

∈

C

1,1

(

Ω

)

, then

l

(

D

u

)

2

1

)

u

D

(

h

3

2

)

y

,

x

(

S

)

y

,

x

(

u

−

≤

ω

1,0

+

ω

0,2

If

u

∈

C

2,2

(

Ω

)

, then

l

(

D

u

)

4

1

)

u

D

(

h

046

,

0

)

y

,

x

(

S

)

y

,

x

(

u

−

≤

2

ω

2,0

+

2

ω

0,2

Proof. We suppose that

u

∈

C

1,1

(

Ω

).

If we use the form (3.1.) of

S

_i_,_j

(

x

,

y

)

and apply the lemma 3.1. for u(x,yj) and u(x,yj+1) then we obtain for (x,y)

∈

Ωi,j.

.

t

)

u

l

u

(

)

t

1

)(

u

l

u

(

v

)

v

1

(

)]

y

,

x

(

u

)

y

,

x

(

u

[

v

)]

y

,

x

(

u

)

y

,

x

(

u

)[

v

1

(

)

u

D

(

h

3

2

)

v

1

(

)

y

,

x

(

u

)

y

,

x

(

S

) 1 , 0 (

1 i j j , 1 i 1 j , 1 i )

1 , 0 (

j , i j j , i 1 j , i

1 j j

0 , 1 j

, i

+ + + + +

+

−

+

−

+

−

+

−

+

ω

+

−

≤

−

(6)

].

,

[

,

),

(

2

1

)

(

3

2

)

(

)

1

(

2

)

(

3

2

)

,

(

)

1

)(

)

,

(

)

1

(

)

,

(

)

1

(

)

,

(

)

1

(

)

(

3

2

)

,

(

)

,

(

1

1 , 0 0

, 1 1

, 0

0 , 1 )

1 , 0 (

, 1 1

1 , 0 )

1 , 0 (

, 1

, 0

1 , 0 1

, 0 0

, 1 ,

+

+ +

∈

+

≤

−

+

≤

−

+

−

+

−

+

−

+

≤

−

j j

j i i

j

j j

j i

y

u

D

l

u

D

h

u

D

l

v

u

D

h

t

u

x

u

D

t

u

x

u

D

vl

v

x

u

D

v

l

x

u

vD

v

l

u

D

h

y

x

u

y

x

S

ζ

η

ω

ζ

η

ω

If

u

∈

C

2,2

(

Ω

)

then the above considerations give:

),

(

4

1

)

(

046

,

0

)

,

(

)

,

(

)

1

(

)

(

046

,

0

)

~

(

2

1

)

~

,

(

)

1

(

2

1

)

,

(

)

1

(

2

1

)

,

(

)

,

(

2

1

)

,

(

)

1

(

)

(

046

,

0

)

,

(

)

,

(

2 , 0 2 2 , 0 2 2

, 0 2

0 , 2 2 2

, 0 2 2

, 0 2 1 2 , 0 2

1 1 , 0 2

, 0 2 1

, 0 0

, 2 2 ,

u

D

l

u

D

h

x

u

D

x

u

D

vl

v

u

D

h

x

u

tD

l

x

u

D

t

l

x

u

D

v

l

u

x

u

D

l

x

u

vD

l

y

x

u

D

l

v

u

D

h

y

x

u

y

x

S

i j

j j

j i

ω

η

ω

η

ω

+

≤

−

+

=

−

+

−

+

≤

−

+

where

η

_j

,

η

_j₊₁

,

~

η

,

~

η

~

,

η

,

η

∈

[

y

_j

,

y

_j₊₁

],

and our theorem is proved.

References

1. B. Bialecki, G. Fairweather, K.A. Remington, Fourier methods for piecewise Hermite bicubic orthogonal spline collocation, East-West J. Numer. Math. 2, 1994. 2. Y. L. Lai, A. Hadjidimos, E.N. Houstis, J.R. Rice, General interior Hermite collocation methods for second order elliptic partial differential equations, Appl. Numer. Math. 16, 1994,(183 – 200).

3. Y. L. Lai, A. Hadjidimos, E.N. Houstis, J.R. Rice, On the iterative solution of Hermite collocation equations, SIAM J. Matrix Anal. Appl. 16, 1995, (254 – 277). 4. B. Bialecki, G. Fairweather, K.R. Bennett, Fast direct solvers for picewine Hermite bicubic orthogonal spline collocation equations, SIAM J. Numer. Anal. 29, 1992, (156 – 173).

5. B.T. Kvasov, Yu. S. Zavialov, V.L. Miroginicinko, Methods of spline functions, Nauka, Moscow, 1980.