Anisotropic error bounds of Lagrange interpolation with any order in two and three dimensions

Applied Mathematics and Computation 217 (2011) 9313–9321

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Anisotropic error bounds of Lagrange interpolation with any order in two and three dimensions q Shaochun Chen a, Yanjun Zheng a, Shipeng Mao b,⇑ a

Department of Mathematics, Zhengzhou University, Zhengzhou 450052, China Institute of Computational Mathematics and Science/Engineering Computing, Academy of Mathematics and System Science, Chinese Academy of Sciences, P.O. Box: 2719, Beijing 100190, China

b

a r t i c l e

i n f o

Keywords: Lagrange interpolation Anisotropic error bounds Newton’s interpolation formula

a b s t r a c t In this paper, using the Newton’s formula of Lagrange interpolation, we present a new proof of the anisotropic error bounds for Lagrange interpolation of any order on the triangle, rectangle, tetrahedron and cube in a unified way. Ó 2011 Elsevier Inc. All rights reserved.

1. Introduction It is known that the polynomial interpolations are the foundations of construction the finite elements and the interpolation error estimates play a key role in deriving a priori error estimates of the finite element methods. The main strategy of the traditional interpolation theory is fairly standard, namely, first deriving the estimate on the reference element and then an application of a coordinate transformation between a general element and the reference element, see [11,7] and references therein. For the triangular and rectangular elements in two dimension and the tetrahedral and cubic elements in three dimension, the mapping between a general element and the reference element is an affine mapping, so in the following we call these elements affine elements. The classical error estimates of the polynomial interpolation on the affine elements need the regular [11] or nondegenerate [7] condition, i.e., the ratio of the diameters of the element and the biggest ball contained in the element is uniformly bounded. This condition restricts the applications of the finite elements. It is found (see e.g., [6,15]) a long time ago that this condition is not necessary for some interpolation error estimates. We call the element does not satisfy the regular condition the anisotropic element. Recently, the research of the anisotropic elements is rapidly developed, and there are several different methods dealing with them. Apel and Dobrowolski [3], Apel [4] gave one anisotropic form of the interpolation error on the reference element. They got the anisotropic interpolation error estimates on a general element for some Lagrange and Hermite elements under the maximal angle and coordinate system conditions. The corresponding appeared derivatives are along the coordinate directions. Chen et al. [9,10] extend this method by presenting a simple anisotropic criterion on the reference element and analyzed some nonconforming elements. Acosta [1], Acosta and Duran [2], Duran [12,13] got the anisotropic error estimates for low order Lagrange and R-T interpolations by using of the average property of the interpolation and the appeared derivatives under consideration are along the directions of the element boundary. The different forms of the anisotropic error estimate of the linear triangular Lagrange interpolation are obtained by the decomposition of the transformation matrix between a general element and the reference element in [14] and by Taylor’s expansion in [8]. In this paper, the anisotropic interpolation error estimates of Lagrange interpolations with any order on the affine elements (triangle, rectangle, cube and tetrahedron) are derived in a unified new way. On the reference element the anisotropic error q

This work is supported by NSFC (11071226 and 10590353).

⇑ Corresponding author.

E-mail addresses: [email protected] (S. Chen), [email protected] (Y. Zheng), [email protected] (S. Mao). 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.04.015

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S. Chen et al. / Applied Mathematics and Computation 217 (2011) 9313–9321

estimates of the interpolations are proved by Newton’s formula of the Lagrange interpolation and a special property of the divided difference, which are different from [4]. The appeared derivatives are along the directions of the element boundary (as in [2,13]) and independent length scales in different directions are extracted (as in [4]). No geometry condition of the element is needed for rectangular and cubic elements. The sine of the biggest internal angle of the element and the regular vertex property factor [2] appear explicitly in the triangular and the tetrahedral elements, respectively, then standard arguments will lead to the estimates that depend on the biggest internal angle of the element and the regular vertex property factor. 2. Lagrange interpolation remainder term on reference elements 2.1. The property of the divided difference Let x0 < x1 <    < xm be a uniform partition, d = xi+1  xi, 0 6 i 6 m  1. It is easy to get the following result by inductive method. Lemma 2.1

Z

x2

dt 1

x1

Z

t 1 þd

dt 2   

Z

t1

t m1 þd

gðt m Þdt m ¼

Z

t m1

x1

dt1

Z

t 1 þd

dt 2   

Z

t1

x0

t m1 þd

gðt m þ dÞdtm :

ð2:1Þ

t m1

Let f[x0, . . . , xm] be the usual divided difference (see [5]), then we get the following lemma. Lemma 2.2. Suppose f(x) is sufficiently smooth, then:

f ½ x0 ; . . . ; x m  ¼

1 m m!d

Z

x1

dt 1

x0

Z

t 1 þd

t1

dt2   

Z

t m1 þd

f ðmÞ ðtm Þdt m :

ð2:2Þ

tm1

Proof. We use the inductive method. Rx When m ¼ 1; f ½x0 ; x1  ¼ 1d x01 f 0 ðt 1 Þdt1 , (2.2) is evident. Suppose (2.2) holds for any m P 1, then:

f ½x0 ; . . . ; xmþ1  ¼ ðf ½x1 ;    ; xmþ1   f ½x0 ; . . . ; xm Þ=ðxmþ1  x0 Þ "Z #, Z t1 þd Z t1 þd Z tm1 þd Z x1 Z tm1 þd x2 1 m ¼ dt 1 dt 2    f ðmÞ ðtm Þdt m  dt 1 dt 2    f ðmÞ ðt m Þdt m ðm!d Þ ðm þ 1Þd x1 t1 t m1 t1 t m1 x0 Z tm1 þd Z x1 Z t1 þd  ðmÞ  1 ð2:1Þ ¼ dt dt    f ðt m þ dÞ  f ðmÞ ðt m Þ dtm 1 2 mþ1 t1 t m1 x0 ðm þ 1Þ!d Z tm1 þd Z x1 Z t1 þd Z tm þd 1 ¼ dt 1 dt 2    dt m f ðmþ1Þ ðtmþ1 Þdtmþ1 : mþ1 x0 t1 t m1 tm ðm þ 1Þ!d This completes the proof.

h

Remark 1. Lemma 2.2 is similar to Hermite–Gennochi Theorem 5, Theorem 3.3. Using the inductive method again, we can get: Lemma 2.3. For all 0 6 l 6 m, f[x0, . . . , xm] can be expressed by

f ½x0 ; . . . ; xm  ¼

ml X

ci f ½xi ; . . . ; xiþl ;

ð2:3Þ

i¼0

where ci (0 6 i 6 m  l) is only dependent on l and d. The interpolation polynomial If(x) of f(x) satisfying If(xi) = f(xi)(0 6 i 6 m) can be expressed in the following two forms, where (2.4) is called Lagrange’s formula and (2.5) is called Newton’s formula (see [5]):

If ðxÞ ¼

m X

f ðxi Þpi ðxÞ;

ð2:4Þ

i¼0

where pi(x) (0 6 i 6 m) 2 Pm (the polynomial space of degree less or equal to m) and pi(xj) = dij, 0 6 i, j 6 m:

If ðxÞ ¼

m X i¼0

f ½x0 ; . . . ; xi 

i1 Y j¼0

ðx  xj Þ:

ð2:5Þ

S. Chen et al. / Applied Mathematics and Computation 217 (2011) 9313–9321

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2.2. Rectangular elements b ¼ ½0; 12 ; d ¼ 1=k; k is a positive integer, ^ b Þ, ^i ¼ id; i ¼ 0; . . . ; k: Suppose u ^ ð^ ^Þ 2 Cð K Let the reference element K xi ¼ y x; y ^ of u ^ satisfying bI u ^ ð^ ^j Þ ¼ u ^ ð^ ^j Þð0 6 i; j 6 kÞ has the following expression (cf. [11]): then bi-k-interpolation polynomial bI u xi ; y xi ; y

bI u ^¼

k X k X   ^j p ^i ð^xÞp ^ ^xi ; y ^ j ðy ^Þ; u i¼0

j¼0

b Þ; p ^i ðtÞ 2 P k ð K ^i ð^ ^ i ðy ^l Þ ¼ dil ; 0 6 i; l 6 k: Obviously: where p xl Þ ¼ p

bI u ^¼u ^;

8u^ 2 Q k ;

ð2:6Þ

where Qk is the polynomial space of the degree 6k with respect to each variable. ^ can be expressed as the following Newton’s formula: Similar as one dimension case, bI u

bI u ^¼

k X k X i¼0

^½^x0 ; . . . ; ^xi ; u

^0 ; . . . ; y ^r  y

i1 r1 Y  Y ^y ^s Þ; ^x  ^xj ðy

r¼0

ð2:7Þ

s¼0

j¼0

^ ½^ ^ ;...;y ^r  is the i-order divided difference with respect to ^ ^ where u x0 ; . . . ; ^ x ;y x and r-order divided difference with respect to y Q  i 0 Qr1 ^ ð^ ^Þ; i1 ^ ^ ^ ^ of u x; y ð y  y Þ ¼ 1 for r = 0. x  x ¼ 1 for i = 0 and s j j¼0 s¼0 b Let us consider a simple example before treating the general case. Taking k = 1 and ooI^xu^ as an example, then it is easy to check that the interpolation function can be written as:

bI u ^¼u ^½^x0 ; y ^0  þ u ^ ½^x0 ; y ^0 ; y ^1 ðy ^y ^0 Þ þ u ^ ½^x0 ; ^x1 ; y ^0 ð^x  ^x0 Þ þ u ^ ½^x0 ; ^x1 ; y ^0 ; y ^1 ð^x  ^x0 Þðy ^y ^0 Þ: So

^ obI u ^ ½^x0 ; ^x1 ; y ^0  þ u ^½^x0 ; ^x1 ; y ^0 ; y ^1 ðy ^y ^0 Þ: ¼u o^x Then it can be checked easily with (2.3) and (2.2) that:

^0  ¼ ^ ½^x0 ; ^x1 ; y u

Z

^x1

^ ½^x; y ^0  ou d^x; ^ ox

^x0

and

^0 ; y ^1  ¼ ^ ½^x0 ; ^x1 ; y u

Z

^x1

^x0

^½^x; y ^0 ; y ^1  ou d^x: o^x

Let us consider the general case and set a = (a1, a2), a1 and a2 are non-negative integers, jaj = a1 + a2, then:

b abI u ^¼ D

k X k X

^ ½^x0 ; . . . ; ^xi ; y ^0 ; . . . ; y ^r Gi ð^xÞRr ðy ^Þ; u

ð2:8Þ

i¼a1 r¼a2

where a

d1 Gi ð^xÞ ¼ a d^x 1

! i1 Y   ^x  ^xj ;

a

r1 Y

d2 ^Þ ¼ Rr ð y ^a2 dy

j¼0

! ^y ^s Þ : ðy

ð2:9Þ

s¼0

b ka ;ka , here Q b m;n is a polynomial space of the degrees of ^ ^Þ 2 Q ^ less or equal to m and n, Obviously Gi ð^ xÞ; Rr ðy x and y 1 2 respectively. By (2.3) and (2.2) we have:

^0 ; . . . ; y ^r  ¼ ^ ½^x0 ; . . . ; ^xi ; y u

ia1 r a2 X X s¼0

j¼0

¼

k

  ^s ; . . . ; y ^sþa2 ^ x^j ; . . . ; ^xjþa1 ; y cjs u

a1 þa2 X ia1 r a2 X

a1 !a2 ! 

Z

1 1

þd

"Z

1 1

R ^xjþ1 ^ xj

dt1

^sþ1 y

R t1 þd t1

^xjþ1

dt 1

Z

^xj

^s y

ta a þa P a Pra 1 2 ^ ¼ ka 1!a 2! i where b LðwÞ s¼0 cjs j¼0 1 2

Z

s¼0

j¼0

ta

cjs

ds1

t1 þd

t1

Z

. . . dta1 1

s1 þd

. . . dsa2 1

s1

   dta1 1

R ta1 1 ta

1 1

h þd R ^

Z

sa

2 1

sa

2 1

þd

# ^ ð^x; y ^Þ oa1 þa2 u b au ^ ^ Þ; dy d^x , b Lð D ^a2 o^xa1 oy

ð2:10Þ

i R s þd R s 1 þd ysþ1 ^ y ^ d^ wd x. ds1 s11    dsa2 1 saa21 ^s y 2

^ s, y ^s+1], [sl, sl+d]1 6 l 6 a2  1) are all in [0, 1], the above integraxjþ1 ; ½t l ; tlþd ð1 6 l 6 a1  1Þ, and [y It is easy to see that ½^ xj ; ^ b or a side of K b or a fixed line lies in K b , hence by the Sobolev trace theorems we can obtain: tion is on K

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S. Chen et al. / Applied Mathematics and Computation 217 (2011) 9313–9321

^ Þj 6 ^ckx ^k jb Lðx

W 1;p ðb KÞ

; 2 6 p 6 1:

Substituting (2.10) into (2.8) results:

b abI u b au ^ ¼ Tb ð D ^ Þ; D

ð2:11Þ

where

^Þ ¼ Tb ðx

k X k X

b ^ ÞGi ð^xÞRr ðy ^Þ: Lðx

ð2:12Þ

i¼a1 r¼a2

Obviously:

8 < k Tb ðx ^ Þk

W m;q ðb KÞ

:b ^ ^; T ðxÞ ¼ x

^k 6 ^ckx

W 1;p ðb KÞ

8 m P 0; 1 6 q 6 1; 2 6 p 6 1;

;

ð2:13Þ

^ 2 Q ka1 ;ka2 : 8x

ð2:11Þ ð2:6Þ b av b ðx bðD b av b abI u b au ^ 2 Q ka1 ;ka2 ; 9v ^ , then T ^Þ ¼ T ^. ^ 2 Q k such that D ^¼x ^Þ ¼ D ^¼x ^ ¼ D In fact 8x

2.3. Cubic elements b ¼ ½0; 13 ; d ¼ We can extend the results from rectangular elements to cubic elements in a straightforward way. Let K ^i ¼ ^zj ¼ id; 0 6 i 6 k, then the interpolation polynomial bI u ^ of u ^ ð^ ^; ^zÞ satisfying bI u ^ ð^ ^r ; ^zl Þ ¼ u ^ ð^ ^r ; ^zl Þ; 1 6 i; r; 1=k; ^ xi ¼ y x; y xi ; y xi ; y l 6 k þ 1, has the following expression:

bI u ^¼

k X k X k X i¼0

r¼0

^0 ; . . . ; y ^r ; ^z0 ; . . . ; ^zl  ^½^x0 ; . . . ; ^xi ; y u

i1 r1 l1 Y Y  Y ^y ^s Þ ð^z  ^zt Þ: ^x  ^xj ðy s¼0

j¼0

l¼0

ð2:14Þ

t¼0

Let a = (a1, a2, a3), then:

b abI u b au ^ ¼ Tb ð D ^ Þ; D

ð2:15Þ

here:

^Þ ¼ Tb ðx

k X k X k X

b ^ ÞGi ð^xÞRr ðy ^ÞHl ð^zÞ; L irl ðx

ð2:16Þ

i¼a1 r¼a2 l¼a3

^Þ is as (2.9) and with Gi ð^ xÞ; Rr ðy a

l1 Y

d3 Hl ð^xÞ ¼ a d^x 3 k b ^Þ ¼ L irl ðx

! ð^z  ^zt Þ ;

a1 þa2 þa3 X ia1 r a2 X la3 X

a1 !a2 !a3 !

where

Z Z Z

ð2:17Þ

t¼0

b¼ ^ dX x

Z

s¼0

j¼0

^xjþ1

dt 1

^xj





cjst

Z

ta

1

   dta1 1 ^sþ1 y

ds1 ^s y

1 1

ð2:18Þ

t 1 þd

"Z 1 þd

ta

b; ^ dX x



t¼0

t1

Z

Z Z Z

Z

s1 þd

s1

   dsa2 1

Z

sa

2 1

sa

2 1

þd

Z

^ztþ1

^zt

dq1

Z

q1 þd

q1

   dqa3 1

Z

qa

3 1

þd

!

#

^ d^z dy ^ d^x: x

qa

3 1

b or on a face (or a fixed internal face) of K b or on an edge (or a fixed internal edge) of K b , similar to The above integration is on K the rectangular case, by trace theorem and Cauchy–Schwarz inequality, we have:

8 < k Tb ðx ^ Þk

W m;q ðb KÞ

:b ^ ^; T ðxÞ ¼ x

^k 6 ^ckx

W 1;p ðb KÞ

;

8 m P 0;

3 6 p 6 1;

1 6 q 6 1;

ð2:19Þ

^ 2 Q ka1 ;ka2 ;ka3 : 8x

2.4. Triangular elements b ¼ fð^ ^Þ; ^ ^ P 0; ^ ^ 6 1g; d ¼ 1=k; ^ ^r ¼ rd; 0 6 i þ r 6 k, then the Lagrange interpolation polynomial Let K x; y x P 0; y xþy xi ¼ id; y bI u ^r Þ ¼ u ^ ð^ ^r Þ; 0 6 i þ r 6 k can be expressed by [10]: ^ of degree k of u ^ ð^ ^Þ, satisfying bI u ^ ð^ xi ; y x; y xi ; y

bI u ^¼

k X ki X i¼0

^r ðy ^ r Þp ^i ð^xÞq ^Þ; ^ ð^xi ; y u

r¼0

b Þ; 0 6 i 6 k; p b Þ; 0 6 r 6 k  i; q ^ r ðy ^r ðy ^i ð^ ^i ð^ ^Þ 2 P ki ð K ^s Þ ¼ drs . where p xÞ 2 P i ð K xj Þ ¼ dij ; q

S. Chen et al. / Applied Mathematics and Computation 217 (2011) 9313–9321

9317

^: The following expression is the Newton’s formula of bI u

bI u ^¼

k X ki X i¼0

^0 ; . . . ; y ^r  ^½^x0 ; . . . ; ^xi ; y u

r¼0

i1 r1 Y  Y ^x  ^xj ^y ^s Þ: ðy

ð2:20Þ

s¼0

j¼0

Let a = (a1, a2), then in the same way as in the rectangular case:

b abI u ^¼ D

k X ki X

b au ^0 ; . . . ; y ^r Gi ð^xÞRr ðy ^Þ ¼ Tb ð D ^ ½^x0 ; . . . ; ^xi ; y ^ Þ; u

ð2:21Þ

i¼a1 r¼a2

where

^Þ ¼ Tb ðx

k X ki X

b ^ ÞGi ð^xÞRr ðy ^Þ; L ir ðx

i¼a1 r¼a2

b au b ^0 ; . . . ; y ^r  ¼ ^Þ ¼ u ^ ½^x0 ; . . . ; ^xi ; y L ir ð D

ia1 r a2 X X

Z Z

^¼ d^xdy

Z

^xjþ1

dt 1

^xj



Z

t 1 þd

t1

   dt a1 1

Z

Z Z

ta

1 1

ta

1 1

b au ^ d^xdy ^ D



s¼0

j¼0

and

^cjs

þd

"Z

^sþ1 y

ds1

Z

^s y

s1 þd

s1

   dsa2 1

Z

sa

2 1

þd

# ^ d^x: dy

sa

2 1

Proceeding as before, we have:

8 < k Tb ðx ^ Þk

W m;q ðb KÞ

:b ^ ^; T ðxÞ ¼ x

^k 6 ^ckx

W 1;p ðb KÞ

;

8m P 0;

1 6 q 6 1;

2 6 p 6 1;

ð2:22Þ

^ 2 Pkjaj : 8x

2.5. Tetrahedral element b ¼ fð^ ^; ^zÞ; ^ ^ P 0; ^z P 0; ^ ^ þ ^z 6 1g; d ¼ 1=k; ^ ^r ¼ rd; ^zl ¼ ld; 0 6 i þ r þ l 6 k, then the interpolation Let K x; y x P 0; y xþy xi ¼ id; y ^ of degree k of u ^ ð^ ^; ^zÞ satisfying bI u ^ ð^ ^r ; ^zl Þ ¼ u ^ ð^ ^r ; ^zl Þ; 0 6 i þ r þ l 6 k has the following form: polynomial bI u x; y xi ; y xi ; y

bI u ^¼

k X ki kir X X i¼0

r¼0

^0 ; . . . ; y ^r ; ^z0 ; . . . ; ^zl  ^ ½^x0 ; . . . ; ^xi ; y u

i1 r1 l1 Y Y  Y ^x  ^xj ^y ^s Þ ðy ð^z  ^zt Þ: j¼0

l¼0

s¼0

t¼0

b a Iu ^ can be expressed as (2.21), here: Let a = (a1, a2, a3), it is easy to see that D

^Þ ¼ Tb ðx

k X ki kir X X

b ^ ÞGi ð^xÞRr ðy ^ÞHl ð^zÞ; L irl ðx

ð2:23Þ

i¼a1 r¼a2 l¼a3

^ Þ is as (2.18) and Gi ð^ ^Þ; Hl ð^zÞ are as (2.9) and (2.17). where b L irl ðx xÞ; Rr ðy Similar to (2.19), we have:

8 < k Tb ðx ^ Þk

W m;q ðb KÞ

:b ^ ^; T ðxÞ ¼ x

^k 6 ^ckx

W 1;p ðb KÞ

;

8 m P 0;

3 6 p 6 1;

1 6 q 6 1;

ð2:24Þ

^ 2 Q ka1 ;ka2 ;ka3 : 8x

2.6. Interpolation remainders b be the reference element ð K b ¼ ½0; 1n ; n ¼ 2 for rectangular element and n = 3 for cubic element; and Let K b ¼ fð^ b ¼ fð^ ^Þ; ^ ^ P 0; ^ ^ 6 1g for triangular element and K ^; ^zÞ; ^ ^ P 0; ^z P 0; ^ ^ þ ^z 6 1g for tetraheK x; y x P 0; y xþy x; y x P 0; y xþy bÞ ! P b is the above Lagrange interpolation operator, here P b ¼ Q k for rectangular and cubic element, dron element) bI : C 0 ð K b ¼ Pk for triangular and tetrahedral element. Then we have: P b Þ,!W m;q ð K b Þ; W kþ1;p ð K b Þ,!C 0 ð K b Þ; a is an index, jaj = m, Theorem 2.4. Suppose that n 6 p 6 1; 1 6 q 6 1; 0 6 m 6 k; W kþ1;p ð K then there exists a constant ^c > 0 such that:

     b a ^ b^   b a ^ u kþ1m;p b :  D ðu  I uÞ q b 6 ^c D L ðK Þ W ðK Þ Proof. By (2.11), (2.15) and (2.20) we have:

  ba u b au b au ^  bI u ^ ¼D ^  Tb D ^ : D Then (2.24) is followed by (2.13), (2.19), (2.22) and (2.24). h

ð2:25Þ

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S. Chen et al. / Applied Mathematics and Computation 217 (2011) 9313–9321

Remark 2. Apel has proved (2.25) in [4], but our method is different from Apel’s. Our main arguments are using the Newton’s formula of Lagrange interpolation and the special property of the divided difference, which admit us to prove (2.25) for rectangular, cubic, triangular and tetrahedral elements in a unified way.

3. Lagrange interpolation errors in general elements 3.1. Rectangular and cubic elements We denote a general rectangular element by K = [x0, x0 + h1]  [y0, y0 + h2] and a general cubic element by [x0, x0 + h1]  [y0, y0 + h2]  [z0, z0 + h3], here a0(x0,y0) or a0(x0, y0, z0) is a vertex of K and h1, h2 or h1, h2, h3 are edge lengths b be the affine mapping from K b to K, then: of K. Let X ¼ F K ð XÞ

b þ a0 ; X ¼ BX

ð3:1Þ

where B = diag(h1, h2) for the rectangular element and B = diag(h1, h2, h3) for the cubic element. Obviously: a b a Da ¼ h D ;

a

b a ¼ ha D a ; D

a1 a2

a

ð3:2Þ

a1 a2 a3

where h ¼ h1 h2 or h ¼ h1 h2 h3 . Lemma 3.1. Suppose ai P 0, 1 6 i 6 N, q P 1, 1/q + 1/q0 = 1, then:

N

q10

N X

ai 6

i¼1

N X

!1q aqi

1

6 Nq

i¼1

N X

ai :

ð3:3Þ

i¼1

b ^  F 1 Let Iu ¼ bI u K ðXÞ; I is defined by (2.7) or (2.14), then I is affine equivalent [11]. Furthermore, we have the following interpolation error estimate. Theorem 3.2. Under the same assumptions as Theorem2.4, then for rectangular and cubic elements, we have:

X

1 1

ju  IujW m;q ðKÞ 6 ^cðdetBÞqp

!1p

p bp  h Db uW kþ1;p ðKÞ

ð3:4Þ

:

jbj¼kþ1m

where ^c is independent of K,det B is the Jacobian of B. Proof

ju  IujW m;q ðKÞ ¼

X a  D ðu  IuÞqq L ðKÞ

!1q

ð3:2Þ

¼

jaj¼m

2 1 q

¼ ^cðdetBÞ 4

X

h

jaj¼m ð3:3Þ

 6 ^cðdetBÞ

1 1

6 ^cðdetBÞqp

1 1 q p

jbj¼kþ1m

X

jaj¼m

jbj¼kþ1m

X

jaj¼m jbj¼kþ1m

aq

   b a ^ b^ q detB D ðu  I uÞ q

h

bp  

D

aþb

!1q

ð2:25Þ

6

^ L ðKÞ

   b aþb ^p  D u p b L ðK Þ

X

X

h

jaj¼m

X

aq

X

p u p

p bp  h Daþb uLp ðKÞ

!1q h

aq

b au ^ jq detB  ^cj D

KÞ W kþ1m;p ðb

jaj¼m

2 !pq 31q X 11 ð3:2Þ 5 ¼ ^cðdetBÞq p 4 jaj¼m

X

X

h

bp  

D

aþb

p u p

L ðKÞ

!pq 31q 5

jbj¼kþ1m

!1p

L ðKÞ

!1p

1 1

¼ ^cðdetBÞqp

X

!1p h jDb ujpW m;p ðKÞ bp



jbj¼kþ1m

3.2. Triangular and tetrahedral elements Let K be a triangle (a tetrahedron) with the vertexes P 0 ; P 1 ; P 2 ðP0 ; P1 ; P2 ; P3 Þ; v 1 ; v 2 ðv 1 ; v 2 ; v 3 Þ be the unit vectors along      edges P0P1, P0P2 (P0P1, P0P2, P0P3) with li = kP0Pik, \P0 be the maximum angle of the triangle K. b The affine mapping F : K ! K is:

b Þ ¼ BX b þ P0 ; X ¼ Fð X

ð3:5Þ

B ¼ B0 K;

ð3:6Þ

where

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S. Chen et al. / Applied Mathematics and Computation 217 (2011) 9313–9321

B0 ¼ ðv 1 ; v 2 ; K ¼ diagðl1 ; l2 Þ for the triangular and B0 ¼ðv 1 ; v 2 ; v 3 Þ; K ¼ diagðl ; l ; l Þ for the tetrahedron. Let    T  T   T

 T     T 1 2 3 T  o o o o o o o o o o ^ ¼ o;o ; r ^ ¼ o;o;o ; ; , by simple compur r ¼ ; ; r ¼ ; ; r ¼ ; ; r ¼ ; ; l l ^ ^ o^z o^ x oy o^ x oy ox oy ox oy oz ov 1 ov 2 ov 1 ov 2 ov 3 tations we have:

^ ¼ KBT r; r 0

rl ¼ BT0 r;

^ ¼ Kr l : r

ð3:7Þ

Let:

^  F 1 Iu ¼ bI u K ðXÞ;

ð3:8Þ

where bI is defined by (2.20) or (2.23). It is well known that (cf. [11]) I is affine equivalent. Theorem 3.3. Under the same assumptions as Theorem 2.4, then for triangular and the tetrahedral elements:

X

1 1

ju  IujW m;q ðKÞ 6 ^cðdetBÞqp l

p bp  l Dbl uW kþ1;p ðKÞ

ojaj a a ov 11 ov 22

ð3:9Þ

;

l

jbj¼kþ1m

where ^c is independent of K, Dal ¼

!1p

ðDal ¼

ojaj a a a Þ; j ov 11 ov 22 ov 33

v jW

m;q ðKÞ l

¼

P

a

jaj¼m kDl

v kqL ðKÞ q

1q

a

a a

a

a a a

; l ¼ l11 l22 ðl ¼ l11 l22 l33 Þ.

Proof

X a  D ðu  IuÞqq l L ðKÞ

ju  IujW m;q ðKÞ ¼ l

!1q

X

1

ð3:7Þ

¼ ðdetBÞq

jaj¼m

ð2:25Þ

 6 ^cðdetBÞ

X

1 q

l

jaj¼m

aq

!1q b au ^j jD

jaj¼m ð3:3Þ

1 1

 6 ^cðdetBÞqp

q

X

jaj¼m

jbj¼kþ1m

l

bp 

2 ð3:7Þ

¼ ^cðdetBÞ

W kþ1m;p ðb KÞ

X

X

1 1

6 ^cðdetBÞqp

 q ^  bI u ^ Þ  D ðu qb L ðK Þ

aq  b a

l

4

X

jaj¼m

p bp  l Dlaþb uLp ðKÞ

p Dbl uW m;p ðKÞ

11 q p

l

!1q

aq

X

l

ðaþbÞp  

aþb

Dl

p u p

L ðKÞ

!qp 31q 5

jbj¼kþ1m

!1p

!1p :



l

jbj¼kþ1m

From (3.7) we can get Lemma 3.4

 m   m;q jv jW m;q ðKÞ 6 BT 0  jv jW ðKÞ ;

ð3:10Þ

l

    where BT 0  is the matrix norm. From Theorem 3.2 and Lemma 3.4, we get: Theorem 3.5. Under the same assumptions as Theorem 2.4, then for triangular and the tetrahedral elements:

ju  IujW m;q ðKÞ

 m 11   q p 6 ^cBT 0  ðdetBÞ

X

l

bp 

p Dbl uW m;p ðKÞ

!1p ð3:11Þ

;

l

jbj¼kþ1m

where ^c is independent of K.  m   Now we estimate BT 0  : (1) Triangular element Let v i ¼ ðcos ui ; sin ui ÞT ; ui be the angle between v i and x-axis, 1 6 i 6 2, and h be the angle between v 1 and v 2 then: 







detB0 ¼ sinðu2  u1 Þ ¼ sin h;       T   T  ð v ; v Þ B0  ¼  ¼  1 2 





1   sin u2 jdetB0 j   cos u2

  sin u1  6 2 :  sin h cos u 1

ð3:12Þ

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S. Chen et al. / Applied Mathematics and Computation 217 (2011) 9313–9321

Fig. 1. The tetrahedron generated by the unit vectors.

(2) Tetrahedral element: Similarly we have:

   T  B0  6

6 : jdetB0 j

ð3:13Þ

Let J ¼ fJ h gh!0 be a family of decompositions of X into tetrahedra. There are three geometry conditions for tetrahedron elements. 1. Regular condition (cf. [11]). J is said to be regular if there exists a constant c0 > 0 such that for any J h 2 J and any K 2 J h we have:

c0 hK 6 qK ; where hK = diam K and qK = diam SK, here SK is the biggest ball contained in K. 2. Regular vertex property (cf. [2]). J is said to have the regular vertex property if there exists a constant c1 > 0 such that for any J h 2 J and any K 2 J h ; K has a vertex P0 such that:

jdetB0 j P c1 ; where B0 see (3.6). 3. Maximum angle condition (cf. [16]). J is said to satisfy the maximum angle condition if there exists a constant 0 < u0 < p such that for any J h 2 J and any K 2 J h , the angles inside the faces and the angles between faces are all bounded above by u0. It is well known that (see [16]) the regular vertex property is stronger than the maximum angle condition and weaker than the regular condition. Obviously detB0 in (3.11) is corresponding to the regular vertex property which can degenerate to flat or needle meshes. Meanwhile detB0 can be expressed by the angles of K at P0. In fact let T = P0Q1Q2Q3 be tetrahedron generated by the unit vectors v i ; 1 6 i 6 3 (see Fig. 1), jTj be the volume of T. Obviously the angles inside the faces and interfacial angles of T are  also K’s ones at P0. Denote by H = jQ3Oj the length of spatial altitude perpendicular to v 1 and v 2 . Denote by r1 = jQ3R1j and r2 = jQ3R2j the alti  tudes perpendicular to v 1 and v 2 , respectively. Then u1 = \Q3R1O,u2 = \Q3R2O are the interfacial angles between faces   P0Q3Q1 and P0Q2Q1, P0Q2Q3 and P0Q2Q1, respectively. Let a0 = \Q1P0Q2,ai = \QiP0Q3, i = 1, 2. Then:

jdetB0 j ¼ 6jTj ¼ 2jMP 0 Q 1 Q 2 j  H ¼ j v 1 jj v 2 j sin a0  r i sin ui ¼ sin a0  j v 3 j sin ai  sin ui ¼ sin a0 sin ai sin ui ; 



i ¼ 1; 2:



Acknowledgment The authors would like to express their sincere thanks to an anonymous referee for his/her many helpful suggestions, together with many corrections of the English and typesetting mistakes. References [1] G. Acosta, Lagrange and average interpolation over 3D anisotropic meshes, J. Comput. Appl. Math. 135 (2001) 91–109.

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