BERNSTEIN-TYPE OPERATORS ON TETRAHEDRONS

STUDIA UNIV. BABEŞ BOLYAI MATHEMATICA Volue LIV Nuber 4 Deceber 2009 BERNSTEIN-TYPE OPERATORS ON TETRAHEDRONS PETRU BLAGA TEODORA CĂTINAŞ AND GHEORGHE COMAN Abstract. The ai of the paper is to costruct Berstei-type operators o tetrahedro with all straight edges ad o tetrahedro with three curved edges defied by soe give fuctios. We study the iterpolatio properties the approxiatio accuracy degree of exactess precisio set ad the reaider of the correspodig approxiatio forulas. The accuracy is also illustrated by uerical exaples.. Itroductio I soe previous papers were costructed ad applied soe iterpolatio operators o triagle with oe curved edge respectively o tetrahedro with straight edges [ 6 7 8 9 2] as well as Berstei-type operators o triagle with all straight edges respectively o triagle with oe curved edge [4 5]. There were studied the iterpolatio properties ad the accuracy of these operators respectively the reaiders of the correspodig approxiatio forulas. The order of a approxiatio operator P is give by the degree of exactess dexp ad by the precisio set presp. Reid that dex P = r if Pf = f for all f Pr ad there exists g Pr+ such that Pg g where Pr deotes the space of polyoials i variables of global degree at ost r. The precisio set of a approxiatio operator is the set of all ooials for which the approxiatio is exact [2]. Received by the editors: 0.09.2009. 2000 Matheatics Subject Classificatio. 4A354A364A80. Key words ad phrases. Berstei operator product operator Boolea su operator approxiatio accuracy tetrahedro. 3

00z PETRU BLAGA TEODORA CĂTINAŞ AND GHEORGHE COMAN The goal of this paper is to study Berstei-type operators o tetrahedros with straight edges respectively with three curved edges give by fuctios. 2. Berstei-type operators o tetrahedros with straight edges By affie ivariace it is sufficiet to cosider oly the stadard tetrahedro T h with vertices V 0 = 0 0 0 V = h 0 0 V 2 = 0 h 0 ad V 3 = 0 0 h with three edges τ τ 2 τ 3 alog the coordiate axes ad with the edges Γ Γ 2 Γ 3 opposite to the vertex V 0. Also oe deotes by σ 02 σ 03 σ 023 ad σ 23 the tetrahedro faces fro the plaes V 0 V V 2 V 0 V V 3 V 0 V 2 V 3 ad V V 2 V 3 respectively see the left side of Figure. V 3 V 3 τ 3 σ 023 h z0z T 0h zz Γ 2 Γ 3 σ 03 x0h x 0yh y τ V 0 τ 2 T 3 V 0 T 2 σ 02 V V 2 2 V Γ V h yy0 xh x0 x00 0y0 Figure. Tetrahedro with straight edges Let Π i i = 2 3 be the parallel plaes to the tetrahedro faces that itersect the tetrahedro edges i three poits ad T i i = 2 3 be the triagles i which the plaes Π i i = 2 3 itersect the tetrahedro faces respectively see the right side of Figure. 2.. Uivariate operators. O each triagle oe defies two Berstei-type operators. Reark. We shall study i detail oly the Berstei-type operators o the triagle T. For the triagles T 2 ad T 3 there are obtaied aalogous results. 4

BERNSTEIN-TYPE OPERATORS ON TETRAHEDRONS Let us cosider the triagle T see Figure 2. 0h zz xh x zz 0yz h y zyz 00z x0z h z0z Figure 2. Triagle T ad For the uifor partitios x = { i h y z } y z i = 0 { y = x j h x z } z j = 0 of the itervals [0 y z h y z y z] [x 0 zx h x z z] respectively oe cosiders the Berstei-type operators B xy ad B yx defied by with ad with B xy Fx y z = p i x y z = B yx Fx y z = p i x y zf i h y z y z i x h y z i i x h y z q j x y zf x j h x z z j=0 5

PETRU BLAGA TEODORA CĂTINAŞ AND GHEORGHE COMAN q j x y z = j y h x z where F is a real-valued fuctio defied o T h. Theore 2.. If F : T h R the: i B xy F = F o σ 023 σ 23 B yx F = F o σ 03 σ 23 ; j j y h x z ii dexb xy = dex Byx = ; iii presb xy = { x i y j z k i = 0 ; j k N } presb yx = { x i y j z k } j = 0 ; i k N ; [ iv B xy e 2jk x y z = x 2 xh x y z ] + y j z k [ B yx e i2k x y z = y 2 + Proof. The relatios y h x y z for i = 0 p i 0 y z = 0 for i > 0; for i = p i h y z y z = 0 for i < ; ] x i z k i j k N. respectively iply that 6 for j = 0 q j x 0 z = 0 for j > 0; for j = q j x h x z z = 0 for j < ; B xy F0 y z = F 0 y z B xy Fh y z y z = F h y z y z 2

BERNSTEIN-TYPE OPERATORS ON TETRAHEDRONS ad B yx Fx 0 y = F x 0 z B yx Fx h x z z = F x h x z z i.e. the iterpolatio properties i. Regardig the approxiatio accuracy we have B xy e 000 x y z= B xy e 00 x y z= p i x y z = i i x x i h y z i h y z h y z i i x x =x =x i h y z h y z i i B xy e x x h y z 200 x y z= i 2 i h y z h y z 2 h y z i i x x = + xh y z i h y z h y z i=2 = x2 + xh y z =x 2 + xh x y z B xy e ijk x y z=y j z k B xy e i00 x y z i = 0 2 j k N respectively B yx e 000 x y z = q j x y z = j=0 B yx e 00x y z = y B yx e i2kx y z = y 2 + y h x y z B yx e ijk x y z = x i z k B yx e 0j0 x y z j = 0 2 i k N 2 that are proved i the sae way which iply ii-iv. Let F = B xy F + R xy F be the approxiatio forula geerated by the operator B xy. 7

PETRU BLAGA TEODORA CĂTINAŞ AND GHEORGHE COMAN Theore 2.2. If F y z C [0 h y z] the R xy Fx y z + h 2δ ω F y z;δ y + z h where ω F y z;δ is the odulus of cotiuity of the fuctio F with regard to the variable x. Moreover if δ = / the R xy Fx y z + h ω F y z; 2. 3 Proof. We have R xy Fx y z p i x y z F x y z F i h y z z y Sice we have [ + δ [ + δ p i x y z δ x ih y z ω + F y z;δ p i x y z x i h y z 2 /2 ] ω F y z;δ xh x y z ] ω F y z;δ. [ ] ax xh x y z h2 z [0 h] 4 T 4 R xy Fx y z respectively for δ = / R xy Fx y z + h 2 We also have R yx Fx y z Theore 2.3. If F y z C 2 [0 h] the + h 2δ ω F y z; δ ω F y z; + h ω F x z; 2.. 5 8 R xy x y z Fx y z = xh F 200 ξ y z 2

BERNSTEIN-TYPE OPERATORS ON TETRAHEDRONS for 0 ξ h y z; y z [0 h] ad where R xy Fx y z h 2 8 M 200F 6 M ijk F = ax T h F ijk x y z. Proof. Sice dex B xy = by Peao s kerel theore follows that R xy where the kerel h y z Fx y z = K 200 x y z; sf 200 s y zds K 200 x y z; s = x s + 0 p i x y z i h y z s + does ot chage the sig K 200 x y z; s 0 s [0 h y z]. By ea value theore oe obtais R xy Fx y z = F 200 ξ y z h y z 0 K 200 x y z; sds = xh x y z F 200 ξ y z 0 ξ h y z. 2 Now the iequality of 4 iplies 6. Reark 2. O the sae way it is proved the evaluatios of the reaider i the forula F = B yx F + Ryx F i.e. for F x z C [0 h x z] R yx Fx y z + h 2 ω F x z; 7 respectively for F x z C 2 [0 h] o T h. R yx Fx y z h2 8 M 020F 8 9

PETRU BLAGA TEODORA CĂTINAŞ AND GHEORGHE COMAN 2.2. Product operators. Let P = B xy B yx ad Q = B yx B xy be the products of the operators B xy respectively We have P F x y z = Q F x y z = ad Byx. j=0 j=0 p i x y zq j i h y z y z F i h y z j ih z + iy p i x j h x z F i jh z + jx z q j x y z Theore 2.4. If F is a real-valued fuctio defied o T h the ad o τ 3 σ 23. Proof. Takig ito accout ad 2 oe obtais P F 0 0 z = F 0 0 z z j h x z z. P F = F 9 Q F = F 0 P F h y z y z = F h y z y z respectively Q F 0 0 z = F 0 0 z Q F h y z y z = F h y z y z for all y z [0 h]. For the approxiatio error of the operators P ad Q we have the followig theore. 0

BERNSTEIN-TYPE OPERATORS ON TETRAHEDRONS Theore 2.5. If F z C [0 h] [0 h] the F P F x y z + hω F z; ad o T h. F Q F x y z + hω F z; 2 Proof. We have F P F x y z [ p i x y zq j i h y z δ y z j=0 x ih y z + p i x y zq j i h y z δ y z j=0 ih z + iy y j + p i x y zq j i h y z ] y z ω F z;δ δ 2 j=0 xh x y z + y h x y z + ω F z;δ δ 2. δ 2 As δ xh x y z y h x y z h y z2 4 h x z2 4 o o [0 h y z] [0 h x z] oe obtais F PF x y z h y z δ 2 + h x z δ 2 2 + ω F z;δ δ 2 h 2 + h δ 2 2 + ω F z;δ δ 2. δ Now for δ = / ad δ 2 = / oe obtais. The iequality 2 is proved i the sae way.

PETRU BLAGA TEODORA CĂTINAŞ AND GHEORGHE COMAN 2.3. Boolea su operators. Let ad S := B xy B yx = B xy + B yx B xy B yx 3 T := B yx be the Boolea sus of the operators B xy ad B yx. B xy = B yx + B xy B yx B xy 4 Theore 2.6. If F is a real-valued fuctio defied o T h the S F = F ad T F = F o σ 03 σ 023 σ 23. Proof. We have: B xy F 0 y z = F 0 y z P F 0 y z = B yx F0 y z which iply that S F = F o σ 023 ; B yx Fx 0 z = F x 0 z P F x 0 z = B xy Fx 0 z which iply that S F = F o σ 03; ad B xy F = F Byx F = F P F = F o σ 23 which iply that S F = F o σ 23. Aalogously it is proved that T F = F o σ 03 σ 023 σ 23. Theore 2.7. If F C T h the F S F x y z + h 2 o T h. 2 + + h ω F x z; 2 ω F y z; + + + hω F z;

BERNSTEIN-TYPE OPERATORS ON TETRAHEDRONS Proof. Fro the idetity oe obtais F S F = F Bxy F + F Byx F F P F F S F x y z R xy Fx y z + R yx Fx y z + F P F x y z ad fro 3 5 the proof follows. Reark 3. The sae iequality is obtaied for the error F T F x y z usig istead of the iequality 2. 3. Berstei-type operators o tetrahedros with three curved edges Oe cosiders also the stadard tetrahedro T h with vertices V 0 = 0 0 0 V = h 0 0 V 2 = 0 h 0 ad V 3 = 0 0 h with three straight edges τ τ 2 τ 3 alog the coordiate axes ad with three curved edges γ γ 2 γ 3 opposite to the vertex V 0 defied respectively by the oe-to-oe fuctios f i ad g i where g i is the iverse of the fuctio f i i = 2 3. Also oe deotes by s 02 s 03 s 023 ad the tetrahedro faces fro the plaes V 0 V V 2 V 0 V V 3 V 0 V 2 V 3 ad V V 2 V 3 respectively by s 23 the curved faced opposite to the vertex V 0 see the left side of Figure 3 ad by t i i = 2 3 the triagles with oe curved edge i which the plaes Π i i = 2 3 itersect the faces of the tetrahedro T h respectively see left side of Figure 3. Next oe cosiders the particular case whe the face s 23 is o the sphere x 2 + y 2 + z 2 = h 2 i.e. f i u = h 2 u 2 ad g i v = h 2 v 2 i = 2 3 see right side of Figure 3 3.. Uivariate operators. Oe each triagle t i i = 2 3 oe defies two Berstei-type operators. We discuss here oly o the triagle t Figure 4. We have B xy Fx y z = h2 y p i x y zf i 2 z 2 y z 3

PETRU BLAGA TEODORA CĂTINAŞ AND GHEORGHE COMAN V 3 V 3 00z τ 3 h 2 z 2 /2 0z t 0h 2 z 2 /2 z γ 2 s 03 s 023 γ 3 x0h 2 x 2 /2 0yh 2 y 2 /2 τ V 0 τ 2 t 3 V 0 t 2 s 02 V V 2 2 V V x00 0y0 γ xh 2 x 2 /2 0 h 2 y 2 /2 y0 Figure 3. Tetrahedro with three curved edges 0h 2 z 2 /2 z xh 2 x 2 z 2 /2 z 0yz h 2 y 2 z 2 /2 yz 00z x0z h 2 z 2 /2 0z Figure 4. Triagle t ad with B yx Fx y z = p i x y z = i j=0 h2 x q j x y zf x j 2 z 2 z i x h2 y 2 z 2 x h2 y 2 z 2 i 4

BERNSTEIN-TYPE OPERATORS ON TETRAHEDRONS respectively q j x y z = j j y h2 x 2 z 2 where F is a real-valued fuctio defied o T h. theores: x h2 x 2 z 2 j Followig the way used i the Sectio 2 oe ca prove the correspodig Theore 3.. If F : T h R the: i B xy F = F o s 023 s 23 B yx F = F o s 03 s 23 ; ii dexb xy = dex B yx = ; iii presb xy = { x i y j z k i = 0 ; j k N } presb yx = { x i y j z k } j = 0 ; i k N ; [x 2 + x h 2 y 2 z 2 x ] iv B xy e 2jkx y z = x i z k [y 2 + y h 2 x 2 z 2 y ] Let B yx e i2k x y z = F = B xy F + Rxy F be the approxiatio forula geerated by the operator B xy Theore 3.2. If F y z C [0 ] h 2 y 2 z 2 the respectively. x i z k i j k N. R xy Fx y z + h 2δ ω F y z;δ y + z h R xy Fx y z + h ω F y z;. 2 Theore 3.3. If F y z C 2 [0 h] the R xy x h2 y 2 z 2 x Fx y z = F 200 ξ y z 2 for 0 ξ h 2 y 2 z 2 y z [0 h] ad R xy Fx y z h 2 8 M 200F. 5

PETRU BLAGA TEODORA CĂTINAŞ AND GHEORGHE COMAN Reark 4. Aalogous results take place for the reaider i the approxiatio forula F = B yx F + Ryx F. 3.2. Product operators. Let P = B xy B yx ad Q = B yx B xy be the products of the operators B xy P Fx y z = respectively Q Fx y z = ad Byx i.e. h2 y p i x y zq j i 2 z 2 y z h2 y F i 2 z 2 2 i j 2 h 2 z 2 + i 2 y 2 j=0 h2 x p i x j 2 z 2 z q j x y z 2 j F i 2 h 2 z 2 + j 2 x 2 h2 x j 2 z 2 j=0 Theore 3.4. If F : T h R the z z. P F = F ad Q F = F o τ 3 s 23. Theore 3.5. If F z C [0 h] [0 h] the F P Fx y z + hω F z; ad F Q Fx y z + hω F z;. 3.3. Boolea su operators. If S = B xy B yx ad T = B yx B xy are the Boolea sus of the operators B xy Theore 3.6. If F : T h R the ad Byx the we have: 6 S F = F ad T F = F o s 03 s 023 s 23.

BERNSTEIN-TYPE OPERATORS ON TETRAHEDRONS Theore 3.7. If F C T h the F S Fx y z + h 2 ω F y z; + + h ω F x z; 2 + + h ω F z; ad a siilar iequality holds for the error F T F. Refereces [] Barhill R. E. Bledig fuctio iterpolatio: a survey ad soe ew results Nuerische Methode der Approxiatiostheorie Bad 3 L. Collatz et al. eds. ISNM Vol.30 Birkhäuser Verlag Basel 976 pp. 43-89. [2] Barhill R.E. Gregory J.A. Polyoial iterpolatio to boudary data o triagles Math. Cop. 29 975 pp. 726-735. [3] Berardi C. Optial fiite-eleet iterpolatio o curved doais SIAM J. Nuer. Aalysis 26989 o.5 pp. 22-240. [4] Blaga P. Coa Gh. Berstei-type operators o triagle Rev. Aal. Nuér. Théor. Approx. 38 2009 o. pp. 9-2. [5] Blaga P. Cătiaş T. Coa Gh. Berstei-type operators o triagle with oe curved side to appear. [6] Coa Gh. Cătiaş T. Iterpolatio operators o triagle with oe curved side to appear. [7] Coa Gh. Cătiaş T. Iterpolatio operators o tetrahedro with three curved sides to appear. [8] Dautray R. Lios J.-L. Aalyse athéatique et calcul uérique pour les scieces et les techiques Vol. 6: Métodes itégrales et uériques Ch. XII pp. 798-834 Masso Paris 988. [9] Gregory J. A. A bledig fuctio iterpolat for triagles Multivariate Approxiatio D.C. Hadscob Acadeic Press Lodo-New York 978 pp. 279-287. [0] Marshall J. A. Mitchell A. R. A exact boudary techique for iproved accuracy i the fiite eleet ethod J. Ist. Math. Appl. 2 973 pp. 355-362. [] Mitchell A.R. McLeod R. Curved eleets i the fiite eleet ethod Coferece o Nuer. Sol. Diff. Eq. Lecture Notes i Math. 363 Spriger Verlag Berli 974 pp. 89-04. 7

PETRU BLAGA TEODORA CĂTINAŞ AND GHEORGHE COMAN [2] Nielso G. Miiu or iterpolatio i triagles SIAM J. Nuer. Aal. 7980 pp. 44-62. [3] Zláal M. Curved eleets i the fiite eleet ethod SIAM J. Nuer. Aal. 0973 pp. 229-240. Babeş-Bolyai Uiversity Faculty of Matheatics ad Coputer Sciece Departet of Applied Matheatics Cluj-Napoca Roaia E-ail address: blaga@ath.ubbcluj.ro Babeş-Bolyai Uiversity Faculty of Matheatics ad Coputer Sciece Departet of Applied Matheatics Cluj-Napoca Roaia E-ail address: tcatias@ath.ubbcluj.ro Babeş-Bolyai Uiversity Faculty of Matheatics ad Coputer Sciece Departet of Applied Matheatics Cluj-Napoca Roaia E-ail address: ghcoa@ath.ubbcluj.ro 8