Department of Applied Mathematics, Tsinghua University Beijing , People's Republic of China Received 17 August 1998; accepted 10 December 1998

Transcription

1 Cmput. Methds Appl. Mech. Engrg. 79 (999) 345±360 The dscrete art cal bndary cndtn n a plygnal art cal bndary fr the exterr prblem f Pssn equatn by usng the drect methd f lnes q Hde Han *, Wezhu Ba Department f Appled Mathematcs, Tsnghua Unversty Bejng 00084, Peple's Republc f Chna Receved 7 August 998; accepted 0 December 998 Abstract The numercal smulatn fr the exterr prblem f Pssn equatn s cnsdered. We ntrduced a plygnal art cal bndary C e and desgned a dscrete art cal bndary cndtn n t by usng the drect methd f lnes. Then the rgnal prblem s reduced t a bndary value prblem de ned n a bnded cmputatnal dman wth a plygnal bndary. The nte element apprxmatn f ths reduced bndary value prblem s cnsdered and t s prved that the nte element apprxmate prblem s well psed. Furthermre numercal examples shw that the dscrete art cal bndary cndtn s very e ectve and mre accurate than the Neumann bndary cndtn whch s ften used n engneerng lteratures. Ó 999 Elsever Scence S.A. All rghts reserved. Keywrds: Pssn equatn; Plygnal art cal bndary; Dscrete art cal bndary cndtn; Fnte element apprxmatn. Intrductn When cmputng the numercal slutn f a bndary value prblem f partal d erental equatns n an unbnded dman, ne ften ntrduces an art cal bndary t cut the unbnded part f the dman and sets up an art cal bndary cndtn at the art cal bndary. Then the rgnal prblem s reduced t a bndary value prblem de ned n a bnded cmputatnal dman. In rder t lmt the cmputatnal cst, the bnded cmputatnal dman must nt be t large. Then the art cal bndary cndtn must be a gd apprxmatn f the exact bndary cndtn n the art cal bndary. Thus the accuracy f the art cal bndary cndtn and the cmputatnal cst are clsely related. Therefre desgnng an art cal bndary cndtn wth hgh accuracy n a gven art cal bndary has becme an e ectve and mprtant methd fr slvng partal d erental equatns n an unbnded dman whch arses n vars elds f engneerng, such as ud w arnd bstacles, cplng f structures wth fndatn, wave prpagatn and s n. There are many authrs wh have wrked n ths subject fr vars prblems by d erent technques. Fr nstance, Engqust and Majda [] desgned absrbng bndary cndtns fr the wave equatn. Gldsten [2] presented the exact bndary cndtn and a sequence f ts apprxmatns at an art cal bndary fr Helmhltz-type equatn n wavegudes. Feng [3] prpsed the asympttc radatn q Ths wrk was supprted partly by the Clmbng Prgram f Natnal Key Prject f Fndatn, Dctral Prgram fndatn f Insttutn f Hgher Educatn and the Natnal Natural Scence Fndatn f Chna. Cmputatn was supprted by the State Key Lab. f Scent c and Engneerng Cmputng n Chna. * Crrespndng authr. E-mal address: hanwu@sun.hep.ac.cn (H. Han) /99/$ - see frnt matter Ó 999 Elsever Scence S.A. All rghts reserved. PII: S ( 9 9 )

2 346 H. Han, W. Ba / Cmput. Methds Appl. Mech. Engrg. 79 (999) 345±360 cndtns fr the reduced wave equatn by usng the asympttc apprxmatn f Hankel functns. Han and Wu [4,5] btaned the exact bndary cndtns and a seres f ther apprxmatns at an art cal bndary fr the Laplace equatn and the lnear elastc system. The exact bndary cndtn at an art cal bndary fr partal d erental equatns n an n nte cylnder was prpsed by Hagstrm and Keller [6,7]. Shrtly after, they used ths technque t slve nnlnear prblems. A famly f art cal bndary cndtns fr unsteady Oseen equatns n the velcty pressure frmulatn wth small vscsty was develped by Halpern and Schatzman [8], whch was then appled t unsteady Naver±Stkes equatns. Nataf [9] desgned an pen bndary cndtn fr a steady Oseen equatn n streamfunctn vrtcty frmulatn, whch s appled t vscs ncmpressble ud w arnd a bdy n a at channel wth slp bndary cndtns n the wall. Hagstrm [0,] prpsed asympttc bndary cndtns at an art cal bndary fr the smulatn f tme-dependent ud ws. Han et al. [2] desgned dscrete art cal bndary cndtns fr ncmpressble vscs ws n an n nte channel by usng a fast teratve methd. Han and Ba [3,4] prpsed dscrete art cal bndary cndtns fr ncmpressble vscs ws n a channel by usng the methd f lnes. Han et al. [5] develped art cal bndary cndtns fr the prblem f n nte elastc fndatn. Recently Ben-Prat and Gvl [6] cnsdered an ellptc art- cal bndary fr the Laplace equatn. One can nd mre references n Ref. [7]. Because f the restrctn f the technques they used, many authrs manly cnsder the regular art cal bndares, such as crcumferences, straght lnes r a segment, as art cal bndares n slvng tw-dmensnal prblems. As we knw, t s very easy t mplement dscretzng a bndary value prblem n a bnded dman wth a plygnal bndary by usng the nte element methd [8]. Thus frm the engneerng pnt f vew, t s natural t ntrduce a plygnal bndary as an art cal bndary fr the prblem n an unbnded dman. Thus hw t desgn an art cal bndary cndtn wth hgh accuracy at a gven plygnal art cal bndary becmes an nterestng pen prblem. In ths paper, we prpse a methd t desgn a dscrete art cal bndary cndtn at a gven plygnal art cal bndary fr the exterr prblem f Pssn equatn by usng the drect methd f lnes. Then the prblem s reduced t a bndary value prblem de ned n a bnded cmputatnal dman. Fnte element apprxmatn f the reduced prblem s als cnsdered. Furthermre numercal examples shw that the dscrete art cal bndary cndtn presented n ths paper s very e ectve. 2. The dscrete art cal bndary cndtn at a plygnal art cal bndary Let C be a bnded, smple and clsed curve n R 2 and X be the unbnded dman wth bndary C. We cnsder the fllwng mdel prblem, the exterr prblem f Pssn equatn: Du ˆ f n X; 2: uj C ˆ g; u s bnded when r! ; 2:2 2:3 where g s a gven functn n C, f s a gven functn n X and we assume that ts supprt s cmpact. Ths prblem s de ned n the unbnded dman X. In Ref. [4], ths prblem was cnsdered. They ntrduced a crcumference as an art cal bndary and desgned a seres f art cal bndary cndtns n t. Then the rgnal prblem (2.)±(2.3) was reduced t a seres f bndary value prblems wth dfferent accuracy n a bnded cmputatnal dman. In ths paper we ntrduce a plygnal art cal bndary C e n X, then X s dvded nt tw parts, the bnded part X and the unbnded part X e ˆ X n X (see Fg. ). C e s gven by r ˆ e h 0 6 h 6 2p 2:4 and such that supp f X. If a sutable bndary cndtn f u at C e s gven, then we can cnsder the bndary value prblem n the bnded dman X. The gal f ths sectn s t desgn the dscrete art cal bndary cndtn at the

3 H. Han, W. Ba / Cmput. Methds Appl. Mech. Engrg. 79 (999) 345± Fg.. gven plygnal art cal bndary C e. We nw cnsder the restrctn f u, the slutn f prblem (2.)± (2.3), n X e. Then we have that Du ˆ 0 n X e ; 2:5 uj Ce ˆ u e h ; h u 0 h ; 2:6 u s bnded when r! : 2:7 Snce the value uj Ce s unknwn, the prblem (2.5)±(2.7) cannt be slved ndependently. If uj Ce 2 H =2 C e s gven, then the prblem (2.5)±(2.7) has a unque slutn u and we knw =nj Ce 2 H =2 C e [20], where H a C e dentes the usual Sblv space n C e wth real number a [8]. On the ther hand, fr any gven u 0 2 H =2 C e, we knw that the prblem (2.5)±(2.7) has a unque slutn ~u [20]. By usng the trace therem [9], we knw ~u=nj Ce 2 H =2 C e. Hence f we de ne ~u=nj Ce as an mage f u 0 frm the space H =2 C e t H =2 C e, we btan a bnded peratr K : H =2 C e! H =2 C e, namely n ˆ K uj Ce : 2:8 Ce In fact the cndtn (2.8) s an exact bndary cndtn sats ed by the slutn f the rgnal prblem (2.)±(2.3). Hence the restrctn f the slutn f the prblem (2.)±(2.3) n X sats es Du ˆ f n X ; 2:9 uj C ˆ g; n ˆ K uj Ce : Ce 2:0 2: Unfrtunately the bnded peratr K s unknwn, thus the prblem (2.9)±(2.) cannt be slved drectly. We nw return t the prblem (2.5)±(2.7) under the assumptn uj Ce s gven. As shwn n Fg., we assume that the plygnal art cal bndary C e has n vertexes fa ˆ x ; x 2 ; ˆ ; 2;... ; n g wth x ˆ R cs h x 2 ˆ R sn h 6 6 n : 2:2

4 348 H. Han, W. Ba / Cmput. Methds Appl. Mech. Engrg. 79 (999) 345±360 Furthermre a n dentes the same vertex as a, but wth the ple crdnate R n ˆ R ; h n ˆ h 2p. Fr the ease f expstn, we assume that h ˆ 0. Then the rays fh ˆ h ; 6 6 n g dvde X e nt n parts X e ˆ fx ˆ x ; x 2 j x 2 X e ; h < h < h g 6 6 n : In the fllwng r am s t map the dman X e nt a strp and ts bndary t a segment by a crdnate transfrmatn. Thus n each subdman X e, we ntrduce the mappng x ˆ qe q cs / sn / a x 2 ˆ qe q sn / sn / a h 6 / 6 h 0 6 q < 6 6 n ; h 6 / 6 h 0 6 q < 6 6 n ; 2:3 wth q ˆ x x 2 x x 2 ja a j x x 2 x x 2 q ; 2:4 x x 2 x 2 x 2 2 sn a ˆ x 2 x 2 ja a j x 2 x 2 q ; 2:5 x x 2 x 2 x 2 2 cs a ˆ x x ja a j x x q : 2:6 x x 2 x 2 x 2 2 It s straghtfrward t check that the transfrmatn (2.3) maps X e nt a sem-n nte strp (see Fg. 2): ~X e ˆ f q; / j h < / < h ; 0 < q < g ˆ ; 2;... ; n : Then X e s mapped nt Xe ~ ˆ f q; / j 0 6 / 6 2p; 0 < q < g and C e s mapped nt ~C e ˆ f 0; / j 0 6 / 6 2pg. Mrever, n X e 6 6 n, we have that Fg. 2.

5 H. Han, W. Ba / Cmput. Methds Appl. Mech. Engrg. 79 (999) 345± q ˆ q e q cs / sn / a q e q sn / x sn / a x 2 ; ˆ q e q cs a q e q sn a ; sn 2 / a x sn 2 / a x 2 ˆ q e q sn a x q sn / sn / a ˆ q e q cs a x 2 q cs / sn / a 2 x 2 2 x 2 2 ˆ q 2 e 2q sn 2 2 a sn 2/ sn 2 / a q 2 sn 2 / sn 2 / a 2 2 ; ; 2:7 2:8 sn2 a q 2 sn a 2 sn / sn / a q ; 2:9 ˆ q 2 e 2q cs 2 2 a sn 2/ sn 2 / a q 2 cs2 a q 2 cs a 2 cs / sn / a q cs 2 / sn 2 / a 2 ; 2:20 2 D ˆ 2 2 x 2 x 2 2 ˆ q 2 e 2q 2 q 2 q sn 2 / a 2 q sn2 / a 2 2 ; 2:2 q 2 e2q dx ˆ sn 2 / a dq d/: 2:22 Furthermre we have that n sn a cs a ˆ q Ce x x 2 qˆ0 q n /ˆh sn h x ˆ R e q ctg h n sn h /ˆh x ˆ R e q ctg h cs h x 2 /ˆh a q cs h x 2 /ˆh 2 sn 2 / a qˆ0 h 6 / 6 h : 2:23 ˆ q e q cs h a q sn h a /ˆh : 2:24 /ˆh a q /ˆh : 2:25 In the new crdnate q; /, the prblem (2.5)±(2.7) s reduced t the fllwng dscntnus ce cent prblem n the sem-n nte strp X ~ e. 2 u sn 2 / a q ctg / a 2 u 2 q ctg / a ˆ 0 q

6 350 H. Han, W. Ba / Cmput. Methds Appl. Mech. Engrg. 79 (999) 345±360 h < / < h 6 6 n; 2:26 u q; h ˆ u q; h 0 6 q < 6 6 n ; 2:27 ctg h a q ˆ ctg h q;h a q q;h 0 6 q < ; 6 6 n ; 2:28 uj qˆ0 ˆ u 0 / ; 2:29 u s bnded when q! ; 2:30 where h ˆ h n, a 0 ˆ a n. We ntce that the ce cents f Eq. (2.26) are ndependent n the varable q. Ths fact wll play an mprtant rle n cnsderng the numercal slutn f the prblem (2.26)±(2.30). Let H 0; 2p dente the usual Sblv space n the nterval 0; 2p [9]. Furthermre we suppse that V ˆ v / j v / 2 H 0; 2p ; v 0 ˆ v 2p ; U ˆ u q; / j fr fxed q 2 0; ; u; q ; 2 u q 2 V : 2 Then the dscntnus ce cent prblem (2.26)±(2.30) s equvalent t the fllwng varatnal±d erental prblem: Fnd u q; / 2 U such that d 2 dq 2 A 2 u; v d dq A u; v A 0 u; v ˆ 0 8v 2 V ; 2:3 uj qˆ0 ˆ u 0 / ; 2:32 where u s bnded when q! ; 2:33 A 2 u; v ˆ Xn ˆ A u; v ˆ Xn ˆ Z h u q; / v / d/; h sn 2 / a 2:34 ctg / a q; / v / h Z h u q; / dv / d/; d/ 2:35 Z 2p dv / A 0 u; v ˆ q; / d/: 2:36 0 d/ We nw cnsder a sem-dscrete apprxmatn f the prblem (2.3)±(2.33). Suppse that 0 ˆ / < / 2 < < / M ˆ 2p s a parttn f the nterval 0; 2pŠ, such that each f fh ; ˆ ; 2;... ; n g s a nde f ths parttn, namely fr every h ˆ ; 2;... ; n there s a / j ˆ h. Let h ˆ max 6 j 6 M / j / j and n V h ˆ v h / j v h / 2 V ; v h j /j ;/ j Š 2 P / j ; / j Š ; j ˆ ; 2;... ; M ;

7 H. Han, W. Ba / Cmput. Methds Appl. Mech. Engrg. 79 (999) 345± U h ˆ u h q; / j fr fxed q P 0; u h ; h q ; 2 u h q 2 V 2 h ; where P m / j ; / j Š dentes the space f plynmals f degree nt grsater that m. Then we get the numercal apprxmatn f the prblem (2.3)±(2.33): Fnd u h q; / 2 U h such that d 2 dq 2 A 2 u h ; v h d dq A u h ; v h A 0 u h ; v h ˆ 0 8v h 2 V h ; 2:37 u h j qˆ0 ˆ u 0;h / ; u h s bnded when q! ; 2:38 where u 0;h / 2 V h such that u 0;h / j ˆ u 0 / j fr j ˆ ; 2;... ; M. Assume that fn j / ; j ˆ ; 2;... ; Mg s a bass f the nte dmensnal space V h such that N j / ˆ d j, 6 ; j 6 M. Let N / ˆ N / ; N 2 / ;... ; N M / Š T and U q ˆ u h q; / ; u h q; / 2 ;... ; u h q; / M Š T fr u h q; / 2 U h. Then we have that u h q; / ˆ N / T U q ; 2:39 u 0;h / ˆ N / T U 0 wth U 0 ˆ u 0 / ; u 0 / 2 ;... ; u 0 / M Š T : 2:40 Thus the dscrete varatnal±d erental prblem (2.37) and (2.38) s equvalent t the fllwng bndary value prblem f a system f rdnary d erental equatns: B 2 U 00 q B U 0 q B 0 U q ˆ 0 0 < q < ; 2:4 where U 0 ˆ U 0 ; U q s bnded when q! ; 2:42 B 2 ˆ Xn ˆ Z h h sn 2 / a N / N / T d/; 2:43 B ˆ Xn ˆ Z h h ctg / a N / N 0 / T N 0 / N / T Š d/; 2:44 B 0 ˆ Z 2p 0 N 0 / N 0 / T d/: 2:45 It s straghtfrward t check that B 2 s a pstve de nte symmetrc matrx, B s an antsymmetrc matrx and B 0 s a sem-negatve de nte symmetrc matrx. We use a drect methd fr slvng the prblem (2.4)± (2.42). Let U q ˆ e kq n; 2:46 where k s a cnstant, n 2 C M t be determned. Substtutng Eq. (2.46) nt the Eqs. (2.4) and (2.42), we get the fllwng generalzed egenvalue prblem k 2 B 2 kb B 0 Š n ˆ 0: 2:47 Let g ˆ kn, then the egenvalue prblem (2.47) s equvalent t the fllwng standard egenvalue prblem: 0 I M n ˆ k I M 0 n ; 2:48 B 0 B g 0 B 2 g

8 352 H. Han, W. Ba / Cmput. Methds Appl. Mech. Engrg. 79 (999) 345±360 where I M dentes the M M unt matrx. After slvng the egenvalue prblem (2.48), we btan the egenvalues k j j ˆ ; 2;... ; M wth nn-pstve real part crrespndng t the egenvalues n j j ˆ ; 2; ; M g j and k ˆ 0, n ˆ ; ;... ; T 2 R M, g ˆ 0 2 R M. Partcularly we assume that k j 6 j 6 r are real egenvalues and k j r 6 j 6 M are cmplex egenvalues wth nnzer magnary parts satsfyng k 2l ˆ k r 2l 6 l 6 M=2 2. Wtht lsng the generalty, we suppse that M s an even number. Thus we btan U q ˆ Xr jˆ b j e qkj n j XM=2 jˆr=2 b 2j R e qk 2j n 2j b 2j I e qk 2j n 2j ; 2:49 where R k and I k dente the real and magnary parts f the cmplex number k, respectvely. Thus U q sats es the rdnary equatns n Eq. (2.4) and the bndary cndtn: U q s bnded, when q!. By usng the cndtn U 0 ˆ U 0, we have that U 0 ˆ Xr jˆ b j n j XM=2 jˆr=2 b 2j R n 2j b 2j I n 2j Š: 2:50 Intrducng matrces D q ˆ e qk n ;... ; e qkr n r ; R e qk r 2 n r 2 ; I e qk r 2 n r 2 ;... ; R e qk M n M ; I e qk M n M ; D 0 D 0 ˆ n ;... ; n r ; R n r 2 ; I n r 2 ;... ; R n M ; I n M Š; B ˆ b ; b 2 ;... ; b M Š T : Frm Eq. (2.50) we btan B ˆ D 0 U 0: 2:5 Substtutng Eq. (2.5) nt Eq. (2.49), we have that U q ˆ D q D 0 U 0: 2:52 Fnally we get the sem-dscrete apprxmate slutn f prblem (2.5)±(2.7) fr gven u 0 h : u h q; / ˆ N / T D q D 0 U 0: 2:53 Ntng Eqs. (2.53) and (2.23), we have that h n ˆ q N / T D 0 0 D 0 Ce 2 sn 2 / a N 0 / U T 0 h < / < h ; 6 6 n : 2:54 In fact the equalty (2.54) s a dscrete art cal bndary cndtn fr the exterr prblem f Pssn equatn n the gven plygnal bndary C e, whch s an apprxmatn f the exact bndary cndtn (2.8). 3. The numercal slutn f the prblem (2.)±(2.3) On the bnded cmputatnal dman X, we nw cnsder the numercal slutn f the prblem (2.)±(2.3). As we knw that the restrctn f u, the slutn f the prblem (2.)±(2.3), sats es the bndary value prblem (2.9)±(2.). Let H X dente the usual Sblv space n X [9] and assume that

9 H. Han, W. Ba / Cmput. Methds Appl. Mech. Engrg. 79 (999) 345± T g ˆ fv 2 H X j v j Cˆ gg; T 0 ˆ fv 2 H X j v j Cˆ 0g: Then the bndary value prblem (2.9)±(2.) s equvalent t the fllwng varatnal prblem: Fnd u 2 T g such that wth a u; v b u; v ˆ f v 8v 2 T 0 ; 3: Z a u; v ˆ ru rv dx; X Z b u; v ˆ Ku v ds; C e Z f v ˆ fv dx: 3:4 X Fr the smplcty, we suppse that C s a plygnal lne n R 2. Let T h be a regular trangulatn such that the ndes n the bndary C e are mapped nt the pnts 0; / j, fr j ˆ ; 2;... ; M by the mappng (2.3). Furthermre let T h ˆ fv h 2 C 0 X jv h j K 2 P K ; 8K 2 T h g; T h g ˆ fv h 2 T h jv h d j ˆ g d j ; fr the nde d j 2 C g; T h 0 ˆ fv h 2 T h jv h j C ˆ 0g: Thus we btan the dscrete frm f the prblem (3.): Fnd u h 2 T h g such that a u h ; v h b u h ; v h ˆ f v h 8v h 2 T h 0 : 3:5 Snce the bnded peratr K s unknwn, we cannt slve the prblem (3.5) drectly. But we have had the dscrete art cal bndary cndtn (2.54). Thus fr u h ; v h 2 T h, let Z b h u h ; v h ˆ Xn h q v h N / T D 0 0 D 0 ˆ h 2 sn 2 / a N 0 / U T 0 ds X n ˆ V T 0 ˆ Z h h sn 2 / a N / N / T D 0 0 D 0 3:2 3:3 2 sn 2 / a N / N 0 / T U 0 d/ wth U 0 ˆ u h 0; / ;... ; u h 0; / M Š T and V 0 ˆ v h 0; / ;... ; v h 0; / M Š T. Usng the blnear frm b h u h ; v h nstead f b u h ; v h n the prblem (3.5) we btan the apprxmatn f the rgnal prblem (3.) (say (2.)±(2.3)): 3:6 Fnd u h 2 T h g such that a u h ; v h b h u h ; v h ˆ f v h 8v h 2 T h 0 : 3:7 After slvng the prblem (3.7), the slutn u h 2 Tg h n the bnded cmputatnal dman X. s an apprxmatn f the rgnal prblem (2.)±(2.3)

10 354 H. Han, W. Ba / Cmput. Methds Appl. Mech. Engrg. 79 (999) 345±360 Fr the blnear frm b h u h ; v h, we have that Lemma 3.. The blnear frm b h u h ; v h s bnded and symmetrc n T h T h. Furthermre b h v h ; v h P 0 fr all v h 2 T h. Prf. Fr any gven u h ; v h 2 T h, ntng Eqs. (2.39) and (3. 6), we have that u h j ~ C e ˆ N / T U 0 ; v h j ~ C e ˆ N / T V 0 : 3:8 3:9 On the dman ~ X e, let u h ˆ N / T D q D 0 U 0; 3:0 v h ˆ N / T D q D 0 V 0; Thus we get the cntnusly extensns f u h and v h n the dman X e. A cmputatn shws ZXe ru h rv h dx ˆ Xn ˆ Z Z h 0 h sn 2 / a ˆ sn a h sn / sn / a h q v h sn a sn / sn / a v h h cs a q q v h cs a cs / sn / a v h d/ dq Z ˆ b h u h ; v h Xn Z h 2 u h A 2 0 h q ; v 2 h ˆ b h u h ; v h : cs / sn / a h h A q ; v h A 0 u h ; v h d/ dq 3: 3:2 Hence Z b h u h ; v h ˆ ru h rv h dx ˆ b h v h ; u h X e 8u h ; v h 2 T h ; 3:3 b h v h ; v h ˆ Z jrv h j 2 dx P 0 X e 8v h 2 T h : 3:4 Frm the Lemma 3. t s straght frward t check that the prblem (3.7) s a well-psed prblem. 4. Numercal mplementatn and examples Let X dente the exterr dman f the unt square, namely X ˆ fx ˆ x ; x 2 j jx j > r jx 2 j > g: We cnsder the numercal slutn f the rgnal prblem (2.)±(2.3) wth gven f and g. We take the art cal bndary C e ˆ f x ; x 2 j x ˆ 2; 2 6 x 2 6 2g [ f x ; x 2 j x 2 ˆ 2; 2 6 x 6 2g. Hence X e ˆ f x ; x 2 j jx j > 2 r jx 2 j > 2g and X ˆ X n X e. Snce the slutn f each example, u x ; x 2, s symmetrc abt x 2 axes and antsymmetrc abt x axes, respectvely, the dman f cmputatn s taken t be the part f X lyng n the rst quadrant. The symmetrc and antsymmetrc bndary cndtns are used alng x ˆ 0 and x 2 ˆ 0, respectvely 0; x 2 x ˆ 0 6 x 2 6 2; 4:

11 H. Han, W. Ba / Cmput. Methds Appl. Mech. Engrg. 79 (999) 345± u x ; 0 ˆ 0 6 x 6 2: 4:2 Three meshes were used n the cmputatn. Fg. 3 shws the trangulatn fr mesh A. On each trangle n mesh A, we cnnected the mdpnts f every tw sdes, thus ths trangle was dvded nt fr small trangles. Then we btaned the re ned mesh B. Mesh C was smlarly generated frm mesh B. Lnear nte element was used n r cmputatn. Example. An unbnded membrane wth a square hle. Let f ˆ 0 n X; 4:3 g x ; x 2 ˆ 2 ln x2 x 2 0:5 2 x 2 x n C 2 0:5 2 : 4:4 Then the rgnal prblem (2.)±(2.3) wth the gven f n (4.3) and g n (4.4) has a unque slutn u x ; x 2 : u x ; x 2 ˆ 2 ln x2 x 2 0:5 2 x 2 x n X: 4:5 2 0:5 2 Let u h dente the nte element apprxmatn by usng the dscrete art cal bndary cndtn (2.54). Table shws the maxmum f the errrs u u h ver the mesh pnts fr meshes A, B and C. Furthermre Table 2 shws the errrs f ku u h k 0;2;X, ju u h j ;2;X and ku u h k ;2;X fr meshes A, B and C. Fr cmparsn we als cmpute the nte element apprxmatn u N h wth Neumann bndary cndtn n the art cal bndary C e fr meshes A, B and C. The maxmum f the errr u u N h ver the mesh pnts s gven n Table 3 and errrs f ku u N h k 0;2;X, ju u N h j ;2;X and ku u N h k ;2;X fr meshes A, B and C are gven n Table 4. Furthermre Fg. 4 shws the related errrs ju u h j=juj 00 n the art cal bndary C e. Fr cmparsn Fg. 5 shws the related errrs ju u N h j=juj 00 n the art cal bndary C e. Fg. 6 shws the cntr plt f the apprxmate slutn u h n mesh C. Example 2. A tw-dmensnal deal (ptental) w arnd a square bstacle (see Fg. 7). In ths example, the unknwn u dentes the streamfunctn f the ud. It s a slutn f the fllwng bndary value prblem: Du ˆ 0 n X; 4:6 uj C ˆ 0; 4:7 Fg. 3. Mesh A.

12 356 H. Han, W. Ba / Cmput. Methds Appl. Mech. Engrg. 79 (999) 345±360 Table Maxmum errr f u u h ver mesh pnts Mesh A B C max ju u h j 2: : : Table 2 Errr f u u h Mesh A B C ku u h k 0;2;X : : : ju u h j ;2;X.03 5: : ku u h k ;2;X : :747 0 Table 3 Maxmum errr f u u N h ver mesh pnts Mesh A B C max ju u N h j Table 4 Errr f u u N h Mesh A B C ku u N h k 0;2;X 8: : : ju u N h j ;2;X ku u N h k ;2;X ru! 0; V T when r! : 4:8 we chse V ˆ :0. Fg. 8 shws the cntr plt f the apprxmate streamfunctn u h n the mesh C. Example 3. A tw-dmensnal ncmpressble w arnd a square bstacle. In ths example, the unknwn u als dentes the streamfunctn f the ud. It s a slutn f the fllwng bndary value prblem: Du ˆ f n X; uj C ˆ 0; 4:9 4:0 ru! 0; V T when r! ; 4:

13 H. Han, W. Ba / Cmput. Methds Appl. Mech. Engrg. 79 (999) 345± Fg. 4. Fg. 5.

14 358 H. Han, W. Ba / Cmput. Methds Appl. Mech. Engrg. 79 (999) 345±360 Fg. 6. Fg. 7. where f x s the gven vrtcty f the ud: ( f x ; x 2 ˆ 4x 2 3x 2 4 x x x x 2 X ; 0 x 2 X e : 4:2 we chse V ˆ :0. Fg. 9 shws the cntr plt f the apprxmate streamfunctn u h n the mesh C. The examples shw that the dscrete art cal bndary cndtn presented n ths paper s very e ectve fr Pssn equatn n exterr dman and mre accurate that Neumann bndary cndtn whch s ften used n engneerng lteratures. Furthermre ths apprach can be appled t slve sme realstc physcs prblems, such as cmputatn f a membrane, a ptental w, an ncmpressble w r a statc electrmagetc eld n an unbnded dman.

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