Диссертация (Некоторые методы проекционного типа численного решения одного класса слабо сингулярных интегральных уравнений), страница 8

Íàïðèìåð, ïðåäîáóñëîâëèâàòåëü Ò. ×åíà ìèíèìèçèðóåò íîðìóÔðîáåíèóñà kCn − An kF è îïðåäåëÿåòñÿ ïðîñòîé ôîðìóëîéck =3.3×èñëåííàÿîïåðàòîðîìkn−kak + an−k ,nnðåàëèçàöèÿ0 ≤ k < n.ìåòîäàÃàëåðêèíàñP h = πhÄëÿ ïîíèìàíèÿ àëãîðèòìîâ ðåàëèçàöèè ìåòîäà Ãàëåðêèíà ñ îïåðàòîðàìèπbh , σ h è `h íàì áóäåò ïîëåçíî êðàòêî íàïîìíèòü ïîäõîä, ïðåäëîæåííûé â [67]äëÿ ÷èñëåííîé ðåàëèçàöèè ìåòîäà Ãàëåðêèíàϕh = $0 π h Λϕh + π h fñ îïåðàòîðîì P h = π h .72Ïîñêîëüêó ïðèáëèæåííîå ðåøåíèå èìååò âèä ϕh (τ ) =nPj=1ãäåχhj−1/2xj χhj−1/2 (τ ), õàðàêòåðèñòè÷åñêàÿ ôóíêöèÿ èíòåðâàëà (τj−1 , τj ), òî ñèñòåìàóðàâíåíèé ìåòîäà Ãàëåðêèíàh−1 (Λϕh , χhi−1/2 )L2 (J) = h−1 (f, χhi−1/2 )L2 (J) ,1≤i≤nâ ìàòðè÷íîé ôîðìå çàïèñè ïðèíèìàåò âèä(3.3.1)x = $0 Λn x + b,ãäå x = (x1 , x2 , .

. . , xn ) , b = (b1 , b2 , . . . , bn ) , ïðè÷åì bi = hTTÌàòðèöà Λn , àïïðîêñèìèðóþùàÿ îïåðàòîð Λ èìååò ýëåìåíòû−1(Λn )ij = h(Λχhj−1/2 , χhi−1/2 )L2 (J)−1Rτjf (τ ) dτ .τj−11 τj τi∫ ∫ E1 (|τ − τ 0 |) dτ dτ 0 = a|j−i| .=2h τj−1 τi−11 Rτ∗E1 (s)ehk (s) ds äëÿ 1 ≤ k < n, ãäåÇäåñü a0 =ak =200ehk ñòàíäàðòíûå áàçèñíûå ôóíêöèè - "øàïî÷êè"; ehk (s) = 1 − |s/h − k| äëÿRτ∗E1 (s)eh0 (s) ds,s ∈ (τk − h, τk + h) è ehk (s) = 0 äëÿ s ∈/ (τk − h, τk + h). ßñíî, ÷òî ìàòðèöà Λnÿâëÿåòñÿ ñèììåòðè÷íîé è òåïëèöåâîé.Çàïèøåì ñèñòåìó (3.3.1) â âèäåAn x = b,ãäå An = In − $0 Λn , In åäèíè÷íàÿ ìàòðèöà ðàçìåðà n.Ìàòðèöà An ñèììåòðè÷íàÿ è òåïëèöåâà. Ïîñêîëüêókxk2 = kϕh k2L2 (J) ,(An x, x) = kϕh k2L2 (J) − $0 (Λϕh , ϕh )L2 (J) ,òî èç íåðàâåíñòâ (1.2.9) ñëåäóåò, ÷òî(1 − $0 )kxk2 ≤ (An x, x) ≤ kxk2 ∀ x ∈ Rn .Òàêèì îáðàçîì, ìàòðèöà An ïîëîæèòåëüíî îïðåäåëåííàÿ è äëÿ íåå(3.3.2)731.1 − $0ðåøåíèÿ ñèñòåìûcond2 (An ) ≤Äëÿ(3.3.2)â[67]ïðåäëàãàåòñÿèñïîëüçîâàòüìåòîä CPCG ñ ïðåäîáóñëîâëèâàòåëåì Ñòðåíãà CSn = In − $0 ΛSn èëè ñCSïðåäîáóñëîâëèâàòåëåì ×åíà CCn = In − $0 Λn .

Çäåñü Λn öèðêóëÿíòíàÿìàòðèöà c ýëåìåíòàìè ïåðâîãî ñòîëáöà aSk = ak äëÿ 0 6 k 6 n/2 è aSk = an−käëÿ n/2 < k < n, à ΛCn öèðêóëÿíòíàÿ ìàòðèöà c ýëåìåíòàìè ïåðâîãîn−kak +n [67] ïîêàçàíî, ÷òîñòîëáöà aCk =ïðåäîáóñëîâëåííûõÒàêèìîáðàçîì,kan−k äëÿ 0 6 k < n.nñîáñòâåííûå çíà÷åíèÿ ñåìåéñòâà ñîîòâåòñòâóþùèõìàòðèöìåòîäêëàñòåðèçóþòñÿe n}{ACPCGñâáëèçèïðåäîáóñëîâëèâàòåëåìåäèíèöû.Ñòðåíãàèïðåäîáóñëàâëèâàòåëåì ×åíà îáëàäàåò ñâîéñòâîì ñâåðõëèíåéíîé ñõîäèìîñòè.Ïðè ýòîì êàæäàÿ èòåðàöèÿ ìåòîäà òðåáóåò âûïîëíåíèÿ ëèøü O(n log2 n)àðèôìåòè÷åñêèõ îïåðàöèé.3.4×èñëåííàÿîïåðàòîðîì3.4.1ðåàëèçàöèÿìåòîäàÃàëåðêèíàñPh = πbhÄèñêðåòèçàöèÿ èíòåãðàëüíîãî óðàâíåíèÿÌåòîä Ãàëåðêèíà ñ P h = πbh èìååò âèäϕh = $0 πbh Λϕh + πbh f(3.4.1)è ÿâëÿåòñÿ êëàññè÷åñêèì ìåòîäîì Ãàëåðêèíà ñ èñïîëüçîâàíèåì ïîäïðîñòðàíñòâà êóñî÷íî ëèíåéíûõ ôóíêöèé.

Çäåñühϕ (τ ) =nXxj ehj (τ ),i=0ãäå xj = ϕh (τj ), à ehj êóñî÷íî ëèíåéíûå áàçèñíûå ôóíêöèè "øàïî÷êè".74Ðàâåíñòâî (3.4.1) îçíà÷àåò òðåáîâàíèå îðòîãîíàëüíîñòè â L2 (J) íåâÿçêèrh = $0 Λϕh + f − ϕh ïðîñòðàíñòâó Sbh (J). Ýòî òðåáîâàíèå ýêâèâàëåíòíîñèñòåìå óðàâíåíèé(ϕh , ehi )L2 (J) = $0 (Λϕh , ehi )L2 (J) + (f, ehi )L2 (J) ,0 ≤ i ≤ n,(3.4.2)êîòîðàÿ â ìàòðè÷íîé ôîðìå çàïèñè èìååò âèä(3.4.3)bb n+1 xb n+1 xb = $0 Λb + b,Tb n+1 , Λb n+1 êâàäðàòíûå ìàòðèöû ïîðÿäêà n + 1 ñ ýëåìåíòàìèãäå Ttij = h−1 (ehj , ehi )L2 (J) , λij = h−1 (Λehj , ehi )L2 (J) , 0 ≤ i ≤ n, 0 ≤ j ≤ n,b = (b0 , b1 , ..., bn )T âåêòîð ñb = (x0 , x1 , ..., xn )T - âåêòîð íåèçâåñòíûõ, bxýëåìåíòàìè bi = h−1 (f, ei )L2 (J) , 0 ≤ i ≤ n.b n+1 ñèììåòðè÷íà, òðåõäèàãîíàëüíà èÊàê íåòðóäíî âèäåòü, ìàòðèöà Tèìååò ñëåäóþùèé âèä:1/3 1/6b n+1T=01/6 2/3 1/60···0···1/6 2/3 1/6 · · ·0··· ··· ··· ··· ···00···01/600···0000 00 00 ··· ··· 2/3 1/6 1/6 1/3b n+1 , àïïðîêñèìèðóþùàÿ îïåðàòîð Λ, ñèììåòðè÷íà è èìååòÌàòðèöà Λïîëîæèòåëüíûå ýëåìåíòû1λij =2hZτ∗ Zτ∗0E1 (|τ − τ 0 |)ehj (τ 0 )ehi (τ )dτ 0 dτ.0Ïðè i = 0, 0 ≤ j ≤ n ôîðìóëà (3.4.4) ïðèíèìàåò âèäZτ∗ Zh1 E1 (|τ − τ 0 |)eh0 (τ ) dτ  ehj (τ 0 ) dτ 0 ,λ0j = γj =2h00(3.4.4)75à ïðè j = n, 0 ≤ i ≤ n ïîñëå çàìåíû s = τ∗ − τ , s0 = τ∗ − τ 0 âèäλi,n = γn−iZτ∗ Zh1 E1 (|s − s0 |)eh0 (s0 ) ds0  ehn−i (s)ds.=2h00Òàêèì îáðàçîì, λ0j = λn−j,n = γj äëÿ 0 ≤ j ≤ n.Ïóñòü òåïåðü 1 ≤ i ≤ j < n.

Ñ ïîìîùüþ çàìåíû s = τ − τi , s0 = τ 0 − τiôîðìóëà (3.4.4) ïðèâîäèòñÿ ê âèäóλij = βj−i =12hτZj−i+1 Zhτj−i−1E1 (|s − s0 |)eh0 (s0 )ds0  ehj−i (s)ds.−hÑëåäîâàòåëüíî ýëåìåíòû λij ïðèíèìàþò ïîñòîÿííûå è ðàâíûå βk çíà÷åíèÿäëÿ âñåõ 1 ≤ i ≤ j < n òàêèõ, ÷òî j − i = k .b n+1 èìååò ñëåäóþùèé âèä:Òàêèì îáðàçîì, ìàòðèöà Λb n+1ΛÂû÷èñëåíèÿñ=γ0γ1γ2· · · γn−1γ1β0β1· · · βn−2γ2β1β0· · · βn−3···············γn−1 βn−2 βn−3 · · ·β0γn−1 γn−2 · · ·γ1γnèñïîëüçîâàíèåìôîðìóëûγn γn−1 γn−2 ··· γ1 γ0Ek (t)=0−Ek+1(t)èèíòåãðèðîâàíèÿ ïî ÷àñòÿì äàþò ñëåäóþùèå âûðàæåíèÿ äëÿ êîýôôèöèåíòîâβk è γk :212β0 = E2 (0) − 2 E4 (0) + 3 [3E5 (0) − 4E5 (h) + E5 (2h)],3hh111β1 = E2 (0) + 2 E4 (0) + 3 [−4E5 (0) + 7E5 (h) − 4E5 (2h) + E5 (3h)],6h2h1βk = 3 [E5 (τk−2 )−4E5 (τk−1 )+6E5 (τk )−4E5 (τk+1 )+E5 (τk+2 )] , 2 ≤ k < n−1,2h1111γ0 = E2 (0) − E3 (0) − 2 E4 (h) + 3 [E5 (0) − E5 (h)],32hhh76111γ1 = E2 (0) + 2 [E4 (0) + 2E4 (h) − E4 (2h)] + 3 [−3E5 (0) + 4E5 (h) − E5 (2h)],62h2h1γk = − 2 [E4 (τk−1 ) − 2E4 (τk ) + E4 (τk+1 )]+2h1+ 3 [E5 (τk−2 ) − 3E5 (τk−1 ) + 3E5 (τk ) − E5 (τk+1 )], 2 ≤ k < n,2h111γn = E3 (τn ) − 2 [E4 (τn−1 ) − E4 (τn )] + 3 [E5 (τn−3 ) − 2E5 (τn−2 ) + E5 (τn−1 )].2hh2hÇàïèøåì ñèñòåìó (3.4.3) â âèäå(3.4.5)bb n+1 xb = b,Ab n+1 = Tb n+1 − $0 Λb n+1 ñèììåòðè÷íàÿ ìàòðèöà.ãäå AÇàìåòèì, ÷òî èç (1.2.9) ñëåäóåò, ÷òîb n+1 xb, xb)Rn+1 ≤ kϕh k2L2 (J)(1 − $0 )kϕh k2L2 (J) ≤ (ϕh − $0 Λϕh , ϕh )L2 (J) = (Ab ∈ Rn+1 .

(3.4.6)∀xb n+1 - ïîëîæèòåëüíî îïðåäåëåííàÿ.Ñëåäîâàòåëüíî ìàòðèöà A3.4.2Ðàçîêàéìëåíèå ìàòðèöûb n+1Îáðàòèì òàêæå âíèìàíèå íà òî, ÷òî ïîñëå ðàçîêàéìëåíèÿ ìàòðèöû A(óäàëåíèÿ èç íåå ïåðâîé è ïîñëåäíåé ñòðîê à òàêæå ïåðâîãî è ïîñëåäíåãîñòîëáöîâ)ïîëó÷àåòñÿñàìîñîïðÿæåííàÿïîëîæèòåëüíîîïðåäåëåííàÿòåïëèöåâà ìàòðèöà An−1 = Tn−1 − $0 Λn−1 , ãäå ìàòðèöûTn−12/31/6= 0· · ·01/60···2/3 1/6 · · ·1/6 2/3 · · ···· ··· ······01/60 β1β2 β0 ββ0β10  1, Λn−1 =  β20 β1β0 ··· ··· ······2/3βn−2 βn−3 βn−4· · · βn−2 · · · βn−3 .· · · βn−4 ··· ··· · · · β077òàêæå ïîëîæèòåëüíî îïðåäåëåííûå è òåïëèöåâû.b ïîäâåêòîð x = (x1 , x2 , .

. . , xn−1 )T èÂûäåëèì â âåêòîðå íåèçâåñòíûõ xïåðåïèøåì ñèñòåìó (3.4.3) â ñëåäóþùåì âèäå(1/3 − $0 γ0 )x0 + (v, x)Rn−1 + γn xn = b0 ,(3.4.7)x0 v + An−1 x + xn w = b,(3.4.8)γn x0 + (w, x)Rn−1 + (1/3 − $0 γ0 )xn = bn .(3.4.9)Çäåñüv = (1/6−$0 γ1 , −$0 γ2 , .

. . , −$0 γn−1 )T ,w = (−$0 γn−1 , . . . , −$0 γ2 , 1/6−$0 γ1 )T ,b = (b1 , b2 , . . . , bn−1 )T .Èç (3.4.8) ñëåäóåò, ÷òî−1−1x = A−1n−1 b − x0 An−1 v − xn An−1 w.(3.4.10)Ïîäñòàâëÿÿ ýòî âûðàæåíèå â (3.4.7), (3.4.9), ïðèõîäèì ê ñëåäóþùåé ñèñòåìåîòíîñèòåëüíî x0 è xn :−1(1/3 − $0 γ0 − (v, A−1n−1 v)Rn−1 )x0 + v · x + (γn − (v, An−1 w)Rn−1 )xn == b0 − (v, A−1n−1 b)Rn−1 ,(3.4.11)−1(γn − (w, A−1n−1 v)Rn−1 )x0 + (1/3 − $0 γ0 − (w, An−1 w)Rn−1 )xn == bn − (w, A−1n−1 b)Rn−1 .(3.4.12)−1Çàìåòèì, ÷òî âåêòîðû A−1n−1 w è An−1 v ñîâïàäàþò ñ òî÷íîñòüþ äîèíâåðñíîãî ðàñïîëîæåíèÿ ýëåìåíòîâ, ïîýòîìó äîñòàòî÷íî íàéòè ëèøü îäèíèç íèõ.Òàêèì îáðàçîì, âû÷èñëåíèå ðåøåíèÿ ñèñòåìû (3.4.7) (3.4.9) ñîñòîèò èçñëåäóþùèõ ýòàïîâ.78−11. Âû÷èñëåíèå A−1n−1 b è An−1 v.2. Ðåøåíèå ñèñòåìû (3.4.11), (3.4.12) îòíîñèòåëüíî x0 è xn .3. Âû÷èñëåíèå x ïî ôîðìóëå (3.4.10).ßñíî, ÷òî îñíîâíûå âû÷èñëèòåëüíûå çàòðàòû ïðîèçâîäÿòñÿ íà ýòàïå 1è ñâÿçàíû ñ íåîáõîäèìîñòüþ ðåøåíèÿ ñèñòåì ëèíåéíûõ àëãåáðàè÷åñêèõóðàâíåíèé ñ òåïëèöåâîé ìàòðèöåé An−1 .3.4.3Ïðèìåíåíèå ìåòîäà CPCG ê ðåøåíèþ ñèñòåìû ñ ìàòðèöåéAn−1 = Tn−1 − $0 Λn−1 .

Êëàñòåðèçàöèÿ ñîáñòâåííûõ çíà÷åíèéÏóñòü An−1 = Tn−1 − $0 Λn−1 . Çàìåòèì, ÷òî èç îöåíêè (3.4.6) ñëåäóåò, ÷òî(1 − $0 )kϕh k2L2 (J) ≤ (An−1 x, x)Rn−1 ≤ kϕh k2L2 (J) ∀ x ∈ Rn−1 ,n−1Pãäå ϕh (τ ) =xj ehj (τ ).j=1Òàêèì îáðàçîì(3.4.13)1 − $0 ≤ λmin (An−1 ) ≤ λmax (An−1 ) ≤ 1.Ïðèìåíèì ê ðåøåíèþ ñèñòåìû An−1 x = b ìåòîä CPCG ñ öèðêóëÿíòíûìïðåäîáóñëîâëèâàòåëåì Ò.

×åíàe n−1 − $0 Λe n−1Cn−1 = Te n−1 , Λe n−1çäåñü T öèðêóëÿíòíûå ìàòðèöû c ïåðâûìè ñòîëáöàìè(2/3, et1 , . . . , etn−2 )T , (β0 , βe1 , . . . , βen−2 )T , ãäå et1 = etn−2 =n−2 e, tk = 06(n − 1)n−1−käëÿ 1 < k < n − 2, βek =βk +kβn−1−k äëÿ 1 ≤ k < n − 1.n−1n−1Èçâåñòíî [85], ÷òî äëÿ ïðåäîáóñëîâëèâàòåëÿ Ò. ×åíà ñïðàâåäëèâûíåðàâåíñòâàλmin (An−1 ) ≤ λmin (Cn−1 ) ≤ λmax (Cn−1 ) ≤ λmax (An−1 ).79ïîýòîìó èç (3.4.13) ñëåäóþò îöåíêè1 − $0 ≤ λmin (Cn−1 ) ≤ λmax (Cn ) ≤ 1,kC−1n k2 <1.1 − $0(3.4.14)Ëåììà 3.4.1. Ñïðàâåäëèâà îöåíêà11kAn−1 − Cn−1 kF ≤ Mh = √ + √ 23 23Z∞E12 (t)t dt + h0Zh1/2E12 (t) dt0.(3.4.15)Äîêàçàòåëüñòâî. Êàê íåòðóäíî âèäåòü,2(n − 2)2(n − 2)212ekTn−1 − Tn−1 kF =+<.36(n − 1)2 36(n − 1)218Ïîñêîëüêó βk − βekk(βk − βn−1−k ),n−1=n−1−k(βn−1−k − βk ), òîn−1e n−1 k2 =kΛn−1 − ΛFn−2X(n − 1 − k)(βk − βek )2 +k=1n−2Xβn−1−k − βen−1−k=k(βn−1−k − βen−1−k )2 =k=1=n−2Xk(n − 1 − k)k=1n−12(βk − βn−1−k ) ≤ 2∞Xkβk2 .k=1Çàìåòèì, ÷òî ïðè 1 ≤ k < n − 1 ñïðàâåäëèâà îöåíêà# τ∗ " h2τk R R20hh 002kβk = 3≤E1 (|s − s |)e0 (s) ds ek (s ) ds2h 0 −h"#"#ττhh∗∗RRRRτk≤ 3eh0 (s)2 ds ehk (s0 ) ds0E12 (|s − s0 |) ds ehk (s0 ) ds0 =2h 0 −h0 −hτk Rh Rτ∗ 2=E1 (|s − s0 |) ehk (s0 ) ds0 ds.3h −h 0Âîñïîëüçîâàâøèñü òåì, ÷òî∞Pτk ehk (s0 ) ≡ s0 ïðè s0 ≥ 0, òàê êàê ýòàk=1ôóíêöèÿ ïðåäñòàâëÿåò ñîáîé ëèíåéíóþ èíòåðïîëÿöèþ ôóíêöèè y(s0 ) = s0 ,80èìååì τ∗h R∞RP1e n−1 k2 ≤ 2E12 (|s − s0 |)τk ehk (s0 ) ds0 ds ≤k(βkh )2 ≤kΛn−1 − ΛF3hk=1k=1−h 0R∞ 21 Rh R∞ 21 Rh R∞ 2000≤E (|s − s |)s ds ds =E (|t|)t dt + E1 (|t|) dt s ds +3h −h 0 13h 0 −s 1−sR∞1 Rh R∞ 2+E1 (t)t dt + E12 (t) dt (−s) ds =3h 0 ss!!hh∞s∞RRRRR11=2E12 (t)t dt + E12 (t) dt s ds ≤2 E12 (t)t dt + h E12 (t) dt .3h300 s00∞PÈç äîêàçàííûõ îöåíîê ñëåäóåò, ÷òîe n−1 kF + kΛn−1 − Λe n−1 kF ≤ Mh .kAn−1 − Cn−1 kF ≤ kTn−1 − TËåììà äîêàçàíà.e n−1 êëàñòåðèçóþòñÿ âáëèçèÏîêàæåì, ÷òî ñîáñòâåííûå ÷èñëà ìàòðèö Aåäèíèöû.e n−1 ) ≤Òåîðåìà 3.4.1.