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Алгоритмічні аспекти методу скінченних елементів: Автореф. дис… канд. фіз.-мат. наук / Г.О. Сбродова, НАН України. Ін-т кібернетики ім. В.М.Глушкова

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IAOeIIIAEUeIA AEAAeAIIss IAOE OE?A?IE

IINOEOOO EIAA?IAOEEE IIAII A.I. AEOOEIAA

Ia i?aaao ?oeiieno

Na?iaeiaa Aaeeia Ieaenaiae?iaia

OAeE 519.6

AEAI?EOII*II ANIAEOE IAOIAeO

NEII*AIEO AEAIAIOIA

01.01.07 – ia/enethaaeueia iaoaiaoeea

Aaoi?aoa?ao

aeena?oaoei? ia caeiaoooy a/aiiai nooiaiy

eaiaeeaeaoa oiceei-iaoaiaoe/ieo iaoe

Ee?a – 1998

Aeena?oaoeiy ? ?oeiien.

?iaioa aeeiiaia ia eaoaae?i /enaeueieo iaoiaeia

iaoaiaoe/ii? oiceee Ee?anueeiai oiiaa?neoaoo

?iaii Oa?ana Oaa/aiea.

Iaoeiaee ea??aiee: aeieoi? o?ceei-iaoaiaoe/ieo iaoe,

i?ioani? IAEA?IA Aieiaeeie? Eaii?aeiae/,
Ee?anueeee oi?aa?neoao ?iai? Oa?ana Oaa/aiea, oaeoeueoao
e?aa?iaoeee, caa?aeoaa/ eaoaae?ith /enaeueieo iaoiae?a
 iaoaiaoe/ii? o?ceee

Io?oe?ei? iiiiaioe: aeieoi? o?ceei-iaoaiaoe/ieo iaoe,

i?ioani? I?EEAC*EEIA A?eoi?
Aai?a?eiae/, 

Ee?anueeee oi?aa?neoao ?iai? Oa?ana Oaa/aiea,
oaeoeueoao e?aa?iaoeee, eaoaae?a ia/enethaaeueii?
iaoaiaoeee

eaiaeeaeao o?ceei-iaoaiaoe/ieo iaoe,

ssEIAE?A Ieoaeei Oaaei?iae/, noa?oee
iaoeiaee ni?a?ia?oiee ?inoeoooo e?aa?iaoeee ?iai?
A.I.Aeooeiaa IAI Oe?a?ie

I?ia?aeia onoaiiaa – Euea?anueeee aea?aeaaiee oi?aa?neoao ?iai? Iaaia
O?aiea,

oaeoeueoao i?eeeaaeii? iaoaiaoeee,

eaoaae?a ia/enethaaeueii? iaoaiaoeee

Caoeno aiaeaoaeaoueny “25” a?oaeiy 1998 ?. i 11 aiaeeii ia caniaeaiii
niaoeiaeiciaaii? a/aii? ?aaee Ae 26.194.01 i?e Iinoeoooi eiaa?iaoeee
iiaii A.I.Aeooeiaa IAI Oe?a?ie ca aae?anith: 252022, Ee?a-22, i?iniaeo
Aeaaeaiiea Aeooeiaa, 40.

C aeena?oaoei?th iiaeia iciaeiieoenue a iaoeiai-oaoii/iiio a?oiai
Iinoeoooo eiaa?iaoeee iiaii A.I. Aeooeiaa IAI Oe?a?ie.

Aaoi?aoa?ao ?icineaiee “25” eenoiiaaea 1998 ?.

A/aiee nae?aoa?

niaoeiaeiciaaii? a/aii? ?aaee II?NA?IEI A.A.

CAAAEUeIA OA?AEOA?ENOEEA ?IAIOE

AEOOAEUeIINOUe OAIE

Iaeiei ic iaeaieueo iioe?aieo iaoiaeia ?ica’ycoaaiiy caaea/
iaoaiaoe/ii? oiceee ? iaoiae neii/aiieo aeaiaioia. ?icaeoie oai?i?
oeueiai iaoiaeo ca’ycaiee c iiaiaie oaeeo aiaeiieo iaoaiaoeeia, ye ?.
Eo?aio, E. O?iae?ion, I.A. Naia?nueeie, A.No?aa, Aeae. Iaeai, I.
Aaaooea, A.I. Ia?/oe, N.A. Iioeii, TH.E. Aeai’yiiae/, I.Ceieaae/, A.A.
Ei?i??a oa aaaaoueio iioeo. ?acii c oei ?yae aeai?eoii/ieo aniaeoia
neii/aiiiaeaiaioieo iaoiaeia eeoeeeny aiaee?eoeie, oi/a aiie e
noaiiaeyoue cia/iee iioa?an aeey i?aeoeee. Na?aae ieo iniaeeao ?ieue
aiaeia?athoue ieoaiiy i?i aoaeoeaio ioeiieo noaeeo noieeinoi a ?icieo
ii?iao, ai aiae ieo caeaaeeoue aeai? no?aoaai? o?eaiaoeyoei? iaeanoi,
iaoiiaeaiinoue aiaeiiaiaeieo aeaaa?a?/ieo nenoai, a oaeiae aaee/eia
noaeeo, ui aoiaeyoue a ai?ii?ii ioeiiee oi/iinoi neii/aiiiaeaiaioieo
ai?ieneiaoeie. Cai?iiiiiaaia aeena?oaoeiy i i?enay/aia aeineiaeaeaiith
aeuaaeacaieo ieoaiue. E??i oeueiai, a iie iaa?oioiaaii eiiaiiiaaiee
iaoiae aeey ?ica’ycoaaiiy neii/aiiiaeaiaioii? ai?ieneiaoei? caaea/i i?i
caei ieanoei, yeee aaco?oueny ia ia?oiaiio iaoiaei oa iaoiaei aei/ii?
aa?oiuei? ?aeaenaoei?, c ioeiieith aeai?eoii/ii? neeaaeiinoi
(o?oaeiiinoeinoi), a oaeiae ciaeaeaii noaei a oai?aiao i?i i?iaeiaaeaiiy
eoneiai-iieiiiiiaeueieo ooieoeie. Ana oea iaoiiaeth? aeooaeueiinoue
aeena?oaoeieii? ?iaioe.

IAOITH ?IAIOE ?:

– aoaeoeaia aeai?eoii/ia ?aaeicaoeiy noai iaoiaeo neii/aiieo aeaiaioia
aeey caaea/ oai?i? i?oaeiinoi i ia?oaaiai iaoiaeo;

– iiaoaeiaa i?aaeiaoiiaethaa/a a ioa?aoeieieo iaoiaeao aeey
ciaoiaeaeaiiy ?ica’yceo nenoai neii/aiiiaeaiaioieo ?iaiyiue, ui
ai?ieneiothoue caaea/o Aei?ioea aeey ?iaiyiiy Ioanniia a iaeanoyo
neeaaeii? oi?ie;

– io?eiaiiy aoaeoeaieo ioeiiie noaeeo noieeinoi eo?aioianueeeo
ai?ieneiaoeie a iao?eeao i?inoi?ia L2, W21, a oaeiae aneiioioe/iiai
i?aaenoaaeaiiy noaeeo noieeinoi o aeiaaeeo ae?iaeaeaiiy o?eeooieeia;

– iiaoaeiaa aeecueei? aei iioeiaeueii? o?eaiaoeyoei? ( – o?eaiaoeyoei?)
aeey iaeanoae c a?aieoeath, ui iaeaaeeoue aei ?icieo eeania aeaaeeinoi
(iaia?a?aii-aeeoa?aioeeiaaia, eoneiai-aeaaeea a?aieoey c iaioeaaeie
eooaie, a oaeiae eoneiai-aeaaeea a?aieoey c aioo?ioiiie ioeueiaeie
eooaie noaiaiaaeo ii?yaeeia);

– aeiaaaeaiiy oai?aie i?i i?iaeiaaeaiiy eoneiai-eiiieieo nioeiaeo
ooieoeie.

CAAAEUeIA IAOIAeEEA AeINEIAeAEAIIss.

A ?iaioi canoiniaothoueny iaoiaee iaoaiaoe/ii? oiceee,
ooieoeiiiaeueiiai aiaeico i ia/enethaaeueii? iaoaiaoeee.

IAOEIAA IIAECIA

Cai?iiiiiaaia iiaa no?aoaaiy iiaoaeiae o?eaiaoeyoeie a caeaaeiinoi aiae
aeaaeeinoi a?aieoei, ciaeaeaii aoaeoeaii ioeiiee ia?aiao?ia
o?eaiaoeyoei? i aeanieo /enae eaaae?aoe/ieo oi?i o aeiaaeeo
eo?aioianueeeo neii/aieo aeaiaioia. Iaea?aeaii aoaeoeaii ioeiiee noaeeo
noieeinoi eo?aioianueeeo ai?ieneiaoeie a W21. Io?eiaii aneiioioe/ia
i?aaenoaaeaiiy noaeeo noieeinoi o aeiaaeeo ae?iaeaeaiiy o?eeooieeia.
Aeaii oai?aoe/ia iaa?oiooaaiiy ia?oaaiai iaoiaeo aeey ciaoiaeaeaiiy
?ica’yceo ?icieoeaai? noaie, ca’ycaii? c caaea/ath i?i caei ieanoeiee,
ciaeaeaii ioeiiee ia/enethaaeueii? neeaaeiinoi aiaeiiaiaeiiai aeai?eoio.

I?AEOE*IA OeIIIINOUe

?acoeueoaoe, io?eiaii a aeena?oaoei?, iiaeooue ciaeoe canoinoaaiiy a
oai?aoe/ieo aeineiaeaeaiiyo, iia’ycaieo c iaoiaeii neii/aieo aeaiaioia
aeey caaea/ iaoaiaoe/ii? oiceee, a oaeiae aeey i?aeoe/iiai ?ica’ycoaaiiy
aeayeeo caaea/ iaoaiiee oaa?aeiai aeaoi?iiaaiiai oiea.

AI?IAAOeIss ?IAIOE

?acoeueoaoe aeena?oaoei? iioaeieiaaii a 1 iiiia?aoi? oa 5 noaooyo, a
oaeiae aeiiiaiaeaeeny ia eaoaae?i /enaeueieo iaoiaeia iaoaiaoe/ii?
oiceee oaeoeueoaoo eiaa?iaoeee Ee?anueeiai oiiaa?neoaoo iiaii Oa?ana
Oaa/aiea i ia iaoeiaiio naiiia?i eaai?aoi?i? iaoiaeia ia/eneaiue
Iaoeiai-aeineiaeiiai iinoeoooo iaoaiaoeee i iaoaiiee iiaii aeaaeaiiea
A.I. Nii?iiaa Naieo-Iaoa?ao?anueeiai oiiaa?neoaoo.

IA’?I I NO?OEOO?A ?IAIOE

Aeena?oaoeiy neeaaea?oueny ic anooio, o?ueio aeaa, o?ueio aeiaeaoeia,
caeeth/aiiy oa nieneo eioa?aoo?e. Aeena?oaoeiy iaienaia ia 157 noi?iieao
iaoeiiieniiai oaenoo, iinoeoue 5 iaethieia i 12 oaaeeoeue.

CIINO ?IAIOE

A aeaai I ?icaeyiooee eiiaiiiaaiee iaoiae aeey ?ica’ycoaaiiy
neii/aiiaeaiaioii? ai?ieneiaoei? caaea/i i?i caei ieanoeiee, ui
aaco?oueny ia ia?oaaiio iaoiaei i iaoiaei aei/ii? aa?oiuei? ?aeaenaoei?
oa ? aieueo aoaeoeaiei o ii?iaiyiii c iioeie iaoiaeaie.

?icaeyiaii caaea/o caeia ieanoeee a i?yiieooieeo

u(0) = f(x)
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Aeey ?ica’ycaiiy oei?? caaea/i nei?enoo?iinue ianooiiith ?icieoeaaith
noaiith A.A.Ei?i??aa.

6/h3 [(2/h (-ui-1, j (0) + 2ui, j (0) – ui+1, j (0)) – (ui-1, j (1) +
ui+1, j (1)) +

2/h(-ui, j-1 (0) + 2ui, j (0) – ui, j+1 (0)) – ui, j-1 (2) + ui, j+1
(2))] = f(xi).

1/h2 [(6/h (ui-1, j (0) – ui+1, j (0)) + 2(ui-1, j (1) + 4ui, j (1) +
ui+1, j (1)) +

+ (-ui, j-1 (1) + 2ui, j (1) – ui, j+1 (1))] = 0.

1/h2 [(6/h (ui-1, j (0) – ui, j+1 (0)) + 2(ui, j-1 (2) + 4ui, j (2) +
ui, j+1 (2)) +

+ (-ui-1, j (2) + 2ui, j (2) – ui+1, j (2))] = 0.

Ooo uh(0), uh(1), uh (2) – nioeiai ooieoei?, yei aiaeiiaiaeathoue
cia/aiiyi uh(0), du(0)/do1, du(0)/do2, ui caaeaii a aoceao oi
i?yiieooii? nioee, i?e/iio ui = 0, yeui oi , i= (i1, i2, i3). Nenoaio
?icieoeaaeo ?iaiyiue caieoaii o aeaeyaei

My = f,

aea
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N = AO,

picscalex1000100090000036f0300000300170000000000050000000902000000000400
000002010100050000000102ffffff00040000002e011800050000003102010000000500
00000b0200000000050000000c02200900291200000026060f001a00ffffffff00001000
0000c0ffffffbeffffffc0280000de0800000b00000026060f000c004d61746854797065
0000200209000000fa02000010000000000000002200040000002d010000050000001402
a9063303050000001302a90658140500000014024200ff02050000001302be08ff020500
0000140242007e14050000001302be087e14050000001402c106831a050000001302c106
e9260500000014025b004f1a050000001302a5084f1a0500000014025b000f2705000000
1302a5080f2717000000fb0220ff0000000000009001000000cc0402001054696d657320
4e657720526f6d616e2043797200cc7d040000002d01010008000000320ab70844280100
00006e0008000000320ab7084027010000006e0008000000320ad008b315010000006e00
08000000320ad008af14010000006e0008000000320a98073c1001000000540008000000
320a5d05f70601000000540008000000320a1d03040401000000540017000000fb0280fe
0000000000009001000000cc0402001054696d6573204e657720526f6d616e2043797200
cc7d040000002d01020004000000f001010008000000320a2b082f260100000053000800
0000320a2b08192401000000300008000000320a2b086922010000002e0008000000320a
2b08cb20010000002e0008000000320a2b08bb1e01000000300008000000320a2b08991c
01000000300008000000320a2b08771a01000000300008000000320a2106ad2601000000
2e0008000000320a21060f25010000002e0008000000320a2106ff220100000030000800
0000320a2106dd2001000000300008000000320a2106af1e01000000530008000000320a
2106991c01000000300008000000320a2106771a01000000300008000000320ae103ad26
010000002e0008000000320ae1030f25010000002e0008000000320ae103ff2201000000
300008000000320ae103dd2001000000300008000000320ae103bb1e0100000030000800
0000320ae1038d1c01000000530008000000320ae103771a01000000300008000000320a
a101ad26010000002e0008000000320aa1010f25010000002e0008000000320aa1012f23
010000002e0008000000320aa101dd2001000000300008000000320aa101bb1e01000000
300008000000320aa101991c01000000300008000000320aa1016b1a0100000053000800
0000320ae004ae1701000000420008000000320a4408581201000000440008000000320a
4408650f01000000460008000000320a44083d0d01000000300008000000320a44081b0b
01000000300008000000320a44086b09010000002e0008000000320a4408cd0701000000
2e0008000000320a44082f06010000002e0008000000320a44081f040100000030000800
0000320a09061c14010000002e0008000000320a09060c1201000000300008000000320a
0906ea0f01000000300008000000320a0906c80d01000000300008000000320a0906940b
01000000460008000000320a0906130901000000440008000000320a0906200601000000
460008000000320a0906270301000000300008000000320ac9031c14010000002e000800
0000320ac903fa11010000002e0008000000320ac903ea0f01000000300008000000320a
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0000320a8901d80f010000002e0008000000320a8901c80d01000000300008000000320a
89018e0b01000000300008000000320a89010d0901000000300008000000320a89012006
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c00010000000fb0220ff0000000000009001000000020002001053796d626f6c00020400
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d0082c1501000000b40010000000fb0280fe000000000000900100000002000200105379
6d626f6c0002040000002d01020004000000f001010008000000320ae0040a1901000000
3d0008000000320ae004831601000000d70008000000320ae004ba01010000003d000a00
000026060f000a00ffffffff01000000000010000000fb021000070000000000bc020000
00cc0102022253797374656d00cc040000002d01010004000000f0010200030000000000
0000

picscalex100010009000003830200000300170000000000050000000902000000000400
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00002d01010008000000320a0004362301000000d70008000000320a0004911301000000
3d0008000000320a00045b1101000000d70008000000320a0004aa08010000002d000800
0000320a00043703010000002d0008000000320a0004be01010000003d0017000000fb02
20ff0000000000009001000000cc0402001054696d6573204e657720526f6d616e204379
7200ccbf040000002d01020004000000f001010008000000320a94053022010000003200
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320a9405551001000000320008000000320a5403330c01000000320008000000320a5403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picscalex100010009000003cf0100000300170000000000050000000902000000000400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picscalex100010009000003c30300000300170000000000050000000902000000000400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Aaaaeaii ianooiia niiaenoai iao?eoeue:

?-1 = 0, ?0 = 1,

A ?k = A Pk-1 – C ?k-2, k = 1, …, n,

R-1 = 0, ?0 = 1,

C Ri = A Ri-1 – B Ri-2 i = 1, …, n,
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 (1)
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 (2)
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(3)
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 (4)

Oai?aia 1. Iaoae o – ?ica’ycie nenoaie ?iaiyiue I y = f, oiaei aeey
aoaeue-yeeo j, q, k o?e aaeoi?e yj, yq, yk, 0 j 0 0 > 0, oi aeey aeanieo /enae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?aaaaeeeai aneiioioe/ii i?aaenoaaeaiiy:
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 3/4 + 0(3), 2O = 1/ + 0(). (15)

2) sseui = c, 0, c = const > 0, , oi aeey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?ii aneiioioe/ii i?aaenoaaeaiiy:
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 3/4 n(n + 1)/(n2 + n+1) + 0(3),

(16)
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 (n2 + n + 1)/(n(n + 1)) 1/ + 0().

A aeaai III cai?iiiiiaaii ?icii niiniae iiaoaeiae o?eaiaoeyoei? a
caeaaeiinoi aiae aeaaeeinoi a?aieoei. Ia iiaenoaai ?acoeueoaoia ae?oai?
aeaae ooo io?eiaii aoaeoeaii ioeiiee noaeeo noieeinoi aeey caaaeaieo
niiniaia o?eaiaoeyoei?.

Ia ieiueii R2 aaiaeeoueny o?eaiaoeueiaaia eaaae?aoia aai i?yiieooia
nioea i aoaeo?oueny o?eaiaoeyoeiy a i?ea?aie/iie nioci a caeaaeiinoi
aiae aeaaeeinoi a?aieoei. § 3.1 i § 3.2 aeaia aoaeoeaia ioeiiea cieco
noaeeo i?e cai?iiiiiaaiiio niiniai o?eaiaoeyoei? a caeaaeiinoi aiae
oe?eie aea?aii? i?ea?aie/ii? nioae aeey aeai/i
iaia?a?aii-aeeoa?aioeieiaaii? a?aieoei i aiae eooo , ui iinoeoue
eoneiai-aeaaeeo a?aieoeth. *enaeueii ?acoeueoaoe iaaaaeaii a oaaeeoeyo
1-12 aeiaeaoeo 2. A § 3.3 i?iiiiothoueny niiniae iiaoaeiae o?eaiaoeyoei?
aeey eoneiai-aeaaeei? a?aieoei c aioo?ioiiie ioeueiaeie eooaie
noaiaiaaeo ii?yaeeia. A § 3.4 cai?iiiiiaaiee aeai?eoi iiaoaeiae –
o?eaiaoeyoei?, a aeayeiio naini aeecueei? aei iioeiaeueii?.

Icia/aiiy. Na?aae anieyeeo o?eaiaoeyoeie, iiaoaeiaaieo aeey caaeaii?
iaeanoi, -o?eaiaoeyoei?th iacaaii oaeo, aeey yei? aiaeiioaiiy max1,2/
min1,2 ? iiiiiaeueiei.

Noi?ioeth?ii aeayei c aeiaaaeaieo a aeaai III oaa?aeaeaiue.

sse i ?aiioa, iaoae V0 – iaeiiii?iee iiaei?inoi? nioeiaeo
ooieoeie-noaeeo,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 –
i?aaeeueii o?eaiaoeueiaaiee aaaaoieooiee.

Oai?aia 3. (Aeiaaeie aeai/i iaia?a?aii aeeoa?aioeieiaaii? a?aieoei).
Iaoae A N2. Oiaei aeey nioeiaeo ooieoeie , i?oiaiiaeueieo aei
iiaei?inoi?o V0, ni?aaaaeeeai ia?iaiinoi.
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 (17)

aea (i), i = 1, …, 8 – noaei noieeinoi, ui ia/eneai? a aeena?oaoei?
(aeea. ae. III, § 3.1), a oaeiae a oaaeeoeyo 1-4 aeiaeaoeo ??.

Oai?aia 4. (Eoneiai-aeaaeea a?aieoey c iaioeueiaei eooii).

Iaoae A iaeaaeeoue aei eoneiai-aeaaeei? a?aieoei c iaioeueiaei eooii,
oiaei aeey nioeiaeo ooieoeie , ui i?oiaiiaeueii iiaei?inoi?o V0,
ni?aaaaeeeai ia?iaiinoi
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 (18)

aea (i)1,2, i = 1, …., 8 – noaei noieeinoi, ui iaaaaeaii a aeaai III, §
3.2, a oaeiae a oaaeeoeyo 5-12 aeiaeaoeo 2. Aeaei ?icaeyaea?oueny
a?aieoey c ioeueiaei eooii – ci nooiaiaaeie caaino?aiiyie aeaeyaeo y =
oa, a > 1.

Eaia 5. Oaeaeeinoue ciaioaiiy eooia ii?iaeueii? o?eaiaoeyoei? a ieiei
ioeueiaiai eooa nooiaiy a oa?aeoa?eco?oueny aneiioioeeith

= n h 1-1/ a + …, (19)

aea

2 a n01-1/ a n 2 a n11-1/ a ,

n0, n1 – aeayei noaei.

Eaia 6. Iaoae A – eoneiai-aeaaeea a?aieoey c ioeueiaei eooii, oiaei
ni?aaaaeeeai ia?iaiinoi
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1(h) = 3/4 n h 1-1/ a + 0( h 1-1/ a),

2(h) = 1/c h –(1-1/ a) – 3/4 n h 1-1/ a + 0( h 1-1/ a).

Oai?aia 5. (Eoneiai-aeaaeea a?aieoey c ioeueiaei eooii). Aeey nioeiaeo
ooieoeie , ui i?oiaiiaeueii iiaei?inoi?o V0, ni?aaaaeeeai ia?iaiinoi
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 (20)

Aeiaeaoie ? i?enay/aiee aeineiaeaeaiith neiaeia eoneiai-eiieieo nioeiaeo
ooieoeie i ?o i?iaeiaaeaiue, caaeaiee ia o?eaiaoeyoeiyo
eoneiai-aeaaeeeo iaeanoae. ?yae aiaeiieo oai?ai i?i i?iaeiaaeaiiy
nioeiaeo eoneiai-iieiiiiiaeueieo ooieoeie anoaiiaeaii a ca’yceo c
iiaoaeiaith i aeineiaeaeaiiyi ioa?aoeieieo i?ioeania iaoiaeo neii/aieo
aeaiaioia. Iaoa aeaiiai aeiaeaoeo – ni?inoeoe aeiaaaeaiiy oeeo oai?ai i
yaii ia/eneeoe noaei, ui cono?i/athoueny oai.

Aeiaeaoie ?? iinoeoue oaaeeoei noaeeo noieeinoi, io?eiaieo a ?iaioi, oa
iaethiee. Iaaiai?aiiy /enaeueieo ?acoeueoaoia i no?oeoo?e oaaeeoeue
iinoeoueny a § 3.1, § 3.2 aeaae III.

Aeiaeaoie ??? iinoeoue aaaeaiiy aeieniinoi aeanieo /enae eaaae?aoe/ieo
oi?i, ui cono?i/athoueny a aeaai III.

INIIAII ?ACOEUeOAOE

A aeena?oaoei? io?eiaii ianooiii iniiaii ?acoeueoaoe:

– aoaeoeaia aeai?eoii/ia ?aaeicaoeiy noai iaoiaeo neii/aieo aeaiaioia
aeey caaea/ oai?i? i?oaeiinoi i ia?oaaiai iaoiaeo;

– iiaoaeiaa ia?aaeiaoiiaethaa/a a ioa?aoeieieo iaoiaeao aeey
ciaoiaeaeaiiy ?ica’yceo nenoai neii/aiiaeaiaoieo ?iaiyiue, ui
ai?ieneiothoue caaea/o Aei?ioea aeey ?iaiyiiy Ioanniia a iaeanoyo
neeaaeii? oi?ie;

– aoaeoeaii ioeiiee noaeeo noieeinoi eo?aioianueeeo ai?ieneiaoeie a
iao?eeao i?inoi?ia L2, W21;

– aneiioioe/ia i?aaenoaaeaiiy aeacaieo noaeeo noieeinoi o aeiaaeeo
ae?iaeaeaiiy o?eeooieeia;

– iaoiae iiaoaeiae -o?eaiaoeyoei?, a iaaiiio naini aeecueei? aei
iioeiaeueii?, aeey iaeanoae c a?aieoeath, ui iaeaaeeoue aei ?icieo
eeania aeaaeeinoi (iaia?a?aii aeeoa?aioeieiaaia, eoneiai-aeaaeea
a?aieoey c iaioeueiaeie eooaie, a oaeiae eoneiai-aeaaeea a?aieoey c
aioo?ioiiie ioeueiaeie eooaie nooiaiaaeo ii?yaeeia);

– aaaeaiiy oai?aie i?i i?iaeiaaeaiiy eoneiai-eiiieieo nioeiaeo ooieoeie.

INIIAII IOAEIEAOeI? II OAII AeENA?OAOeI?:

1. Na?iaeiaa A.A. Aeai?eoie/aneea aniaeou iaeioi?uo eiia/ii-yeaiaioiuo
iaoiaeia. – E., 1997. – 155 n.

2. Na?iaeiaa A.A. Ia?oaaue iaoiae aeey naoi/iie caaea/e ecaeaa ieanoeiee
a i?yiioaieueieea // Aanoiee NIaAO, na?ey iaoaiaoeea, iaoaieea,
ano?iiiey. Naieo-Iaoa?ao?a: 1992 , aeai. AEIEOE, ? 32343-A92.

3. Na?iaeiaa A.A. Oai?aia i i?iaeieaeaiee naoi/iuo ooieoeee // Aanoiee
NIaAO, na?ey iaoaiaoeea, iaoaieea, ano?iiiiey. Naieo-Iaoa?ao?a: 1994,
aeai. AEIEOE, ? 446-A94.

4. Na?iaeiaa A.A. I o?oaeiaieinoe au/eneaiee i?e i?eaeeaeaiiii ?aoaiee
caaea/e ia ecaeaa ieanoeiee // Aanoiee NIaAO, na?ey iaoaiaoeea,
iaoaieea, ano?iiiiey. Naieo-Iaoa?ao?a: 1994, aeai. AEIEOE, ? 1069-A9.

5. Iaea?ia A.E., Na?iaeiaa A.A. Eiinoaiou onoie/eainoe aii?ieneiaoeee
Eo?aioa // Au/eneeoaeueiay e i?eeeaaeiay iaoaiaoeea. – 1998. – N 82. –
N. 168-192.

6. Iaea?ia A.E., Na?iaeiaa A.I. Ioe?iee noaeeo no?eeino? aiao?eeao
i?inoi??a L2, W12 // A?niee Aea?aeaaiiai oi?aa?neoaoo “Euea?anueea
iie?oaoi?ea”, 1998. – N 337. – N. 350-353.

Na?iaeiaa A.I. Aeai?eoi?/i? aniaeoe iaoiaeo ne?i/aiieo aeaiaio?a. –
?oeiien.

Aeena?oaoe?y ia caeiaoooy a/aiiai nooiaiy eaiaeeaeaoa
o?ceei-iaoaiaoe/ieo iaoe ca niaoe?aeuei?noth 01.01.07 – ia/enethaaeueia
iaoaiaoeea. – Ee?anueeee oi?aa?neoao ?iai? Oa?ana Oaa/aiea, Ee?a, 1998.

Caoeua?oueny aeai?eoi?/ia ?aae?caoe?y noai iaoiaeo ne?i/aiieo aeaiaio?a
aeey caaea/ iaoaiaoe/ii? o?ceee, ca’ycaieo c iia?aoi?ii Eaieana.
Iiaoaeiaaiee ia?aaeiaoiiaethaa/ a ?oa?aoe?eieo iaoiaeao aeey
ciaoiaeaeaiiy ??oaiiy nenoai ne?i/aiii-aeaiaioieo ??aiyiue,
ai?ieneioth/eo caaea/o Aee?eoea aeey ??aiyiiy Ioanniia a iaeanoyo
neeaaeii? oi?ie. Io?eiai? aoaeoeai? ioe?iee eiinoaio o?eaaeino?
eo?aioianueeeo ai?ieneiaoe?e a iao?eeao i?inoi??a L1, L12, a oaeiae
aneiioioe/ia iiaeaiiy icia/aieo eiinoaio o?eaaeino? o aeiaaeeo
ae?iaeaeaiiy o?eeooiee?a. Iiaoaeiaaia aeecueea aei iioeiaeueii?
o?eaiaoeyoe?? (-o?eaiaoeyoe?y) aeey iaeanoae c ei?aeiiii, ui iaeaaeeoue
aei ??cieo eean?a aeaaeeino?. Aeiaaaeaia oai?aia i?i i?iaeiaaeaiiy
eoneiai-e?i?eieo n?oeiaeo ooieoe?e.

Eeth/ia? neiaa: eiinoaioa o?eaaeino?, eo?aioianueea ai?ieneiaoe?y,
o?eaiaoeyoe?y, ia?aaeiaoiiaethaa/.

Na?iaeiaa A.A. Aeai?eoie/aneea aniaeou iaoiaea eiia/iuo yeaiaioia. –
?oeiienue.

Aeenna?oaoeey ia nieneaiea o/aiie noaiaie eaiaeeaeaoa
oeceei-iaoaiaoe/aneeo iaoe ni niaoeeaeueiinoe 01.01.07 – au/eneeoaeueiay
iaoaiaoeea, Eeaaneee oieaa?neoao eiaie Oa?ana Oaa/aiei, Eeaa, 1998 a.

Caueuaaony aeai?eoie/aneay ?aaeecaoeey noai iaoiaea eiia/iuo yeaiaioia
aeey caaea/ iaoaiaoe/aneie oeceee, naycaiiuo n iia?aoi?ii Eaieana.
Iino?iai i?aaeiaoneaaeeaaoaeue a eoa?aoeeiiiuo iaoiaeao aeey
iaoiaeaeaiey ?aoaiey nenoai eiia/iiyeaiaioiuo o?aaiaiee,
aii?ieneie?othueo caaea/o Aee?eoea aeey o?aaiaiey Ioanniia a iaeanoyo
neiaeiie oi?iu. Iieo/aiu yooaeoeaiua ioeaiee eiinoaio onoie/eainoe
eo?aioianeeo aii?ieneiaoeee a iao?eeao i?ino?ainoa L1, L12, a oaeaea
aneiioioe/aneia i?aaenoaaeaiea oeacaiiuo eiinoaio onoie/eainoe a neo/aa
au?iaeaeaiey o?aoaieueieeia. Iino?iaia aeeceay e iioeiaeueiie
o?eaiaoeyoeey (-o?eaiaoeyoeey) aeey iaeanoae n a?aieoeae,
i?eiaaeeaaeauae e ?aciui eeannai aeaaeeinoe. Aeieacaia oai?aia i
i?iaeieaeaiee eoni/ii-eeiaeiuo naoi/iuo ooieoeee.

Eeth/aaua neiaa: eiinoaioa onoie/eainoe, eo?aioianeay aii?ieneiaoeey,
o?eaiaoeyoeey, i?aaeiaoneaaeeaaoaeue.

Sbrodova G.A. The algorithmic aspects of the method of finite elements.
– Manuscript.

Thesis for the scientific degree of the candidate of
physico-mathematical sciences on the speciality 01.01.07- calculus
mathematics. Taras Shevchenko Kiev University, Kiev, 1998.

The algorithmic realization of schemes of the method of finite elements
for problems of the mathematical physics connected with the Laplace
operator is presented. The predefinder in iterative methods for the
determination of a solution for systems of finite-elements equations is
proposed which approximates the Dirichlet problem for the Poisson
equation in areas of the complicated form. The effective valuations of
the stability constants for the Curant approximations in the metrics of
spaces L2, L12 as well as the asymptotic representation of the indicated
constants of stability in the case of the degeneration of triangles are
obtained. The triangulation close to an optimum (-triangulation) is
constructed for areas with the boundaries belong to different classes of
smoothness. The theorem of the prolongation of piecewise linear grid
functions is proved.

Key words: constant of stability, Curant approximation, triangulation,
predefinder.

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