Geodesic
The substantiation of the method of two-sided approximations for eigenvalues of the second-order elliptic operator
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 4 (2001) no. 1, pp. 61-84 
S. B. Sorokin. The substantiation of the method of two-sided approximations for eigenvalues of the second-order elliptic operator. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 4 (2001) no. 1, pp. 61-84. http://geodesic.mathdoc.fr/item/SJVM_2001_4_1_a6/
@article{SJVM_2001_4_1_a6,
     author = {S. B. Sorokin},
     title = {The substantiation of the method of two-sided approximations for eigenvalues of the second-order elliptic operator},
     journal = {Sibirskij \v{z}urnal vy\v{c}islitelʹnoj matematiki},
     pages = {61--84},
     year = {2001},
     volume = {4},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJVM_2001_4_1_a6/}
}
TY  - JOUR
AU  - S. B. Sorokin
TI  - The substantiation of the method of two-sided approximations for eigenvalues of the second-order elliptic operator
JO  - Sibirskij žurnal vyčislitelʹnoj matematiki
PY  - 2001
SP  - 61
EP  - 84
VL  - 4
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/SJVM_2001_4_1_a6/
LA  - ru
ID  - SJVM_2001_4_1_a6
ER  - 
%0 Journal Article
%A S. B. Sorokin
%T The substantiation of the method of two-sided approximations for eigenvalues of the second-order elliptic operator
%J Sibirskij žurnal vyčislitelʹnoj matematiki
%D 2001
%P 61-84
%V 4
%N 1
%U http://geodesic.mathdoc.fr/item/SJVM_2001_4_1_a6/
%G ru
%F SJVM_2001_4_1_a6

Voir la notice de l'article provenant de la source Math-Net.Ru

In the paper, the method of two-sided approximation for the spectral problems with the second-order elliptic operator based on the well-known fictitious method is substantiated. The expansion of eigenvalues of auxiliary problem in power series by small parameter of continuation irrespective of its sign is proved. Unimprovable error bounds of two-sided approximations are obtained. The conjugate-factorized structure of the problem operator plays the decisive role in obtaining the result. The research is realized on discrete level.

[1] Saulev V. K., “Ob odnom metode avtomatizatsii resheniya kraevykh zadach na bystrodeistvuyuschikh vychislitelnykh mashinakh”, Dokl. AN SSSR, 144:3 (1962), 497–500 | MR

[2] Konovalov A. N., “Metod fiktivnykh oblastei v zadachakh filtratsii dvukhfaznoi neszhimaemoi zhidkosti s uchetom kapillyarnykh sil”, Chislennye metody mekhaniki sploshnoi sredy / RAN. Sib. otd-nie. VTs. — Novosibirsk, 3:5 (1972), 52–68 | MR

[3] Bugrov A. N., “Metod fiktivnykh oblastei dlya uravnenii s chastnymi proizvodnymi ellipticheskogo tipa”, Materialy V Vsesoyuznoi konferentsii “Chislennye metody resheniya zadach teorii uprugosti i plastichnosti”. Chast II (Novosibirsk, 1978), 24–35

[4] Nikolaeva N. I., “Metod fiktivnykh oblastei dlya zadach na sobstvennye znacheniya”, Chislennye metody mekhaniki sploshnoi sredy / RAN. Sib. otd-nie. VTs. — Novosibirsk, 10:6 (1979), 105–112 | MR

[5] Konyukh G. V., “Dvustoronnie otsenki dlya resheniya setochnykh zadach”, Variatsionno-raznostnye metody v zadachakh chislennogo analiza, Novosibirsk, 1988, 163–171 | MR | Zbl

[6] Chertova K. A., “Locally two-sided approximate solutions in parabolic problems”, Bulletin of the Novosibirsk Computing Center. Numerical Analysis. — Novosibirsk, 6 (1994), 37–42

[7] Nikiforovskii S. V., Sorokin S. B., “Dvustoronnie priblizheniya dlya sobstvennykh chisel v metode fiktivnykh oblastei”, Chislennye metody mekhaniki sploshnoi sredy / RAN. Sib. otd-nie. VTs. — Novosibirsk, 1986, 17:5, 110–119

[8] Konovalov A. N., Sorokin S. B., Struktura uravnenii teorii uprugosti. Statika, Preprint / AN SSSR. Sib. otd-nie. VTs; 665, Novosibirsk, 1986 | MR

[9] Sorokin S. B., “Numerical solution of ellipic problems with factorized operators”, Bulletin of the Novosibirsk Computing Center. Numerical Analysis. — Novosibirsk, 5 (1994), 87–103 | Zbl

[10] Konovalov A. N., “Sopryazhenno-faktorizovannye modeli v zadachakh matematicheskoi fiziki”, Sib. zhurn. vychisl. matematiki / RAN. Sib. otd-nie. — Novosibirsk, 1:1 (1998), 1–34 | MR

[11] Voevodin V. V., Kuznetsov Yu. A., Matritsy i vychisleniya, Nauka, M., 1984 | MR | Zbl

[12] Mikhlin S. G., Variatsionnye metody v matematicheskoi fizike, Nauka, M., 1970 | MR | Zbl

[13] Sorokin S. B., “Otsenka tochnosti dvustoronnikh priblizhenii dlya zadachi Shturma–Liuvillya”, Sib. zhurn. vychisl. matematiki / RAN. Sib. otd-nie. — Novosibirsk, 3:1 (2000), 77–94 | MR

[14] Marchuk G. I., Agoshkov V. I., Shutyaev V. P., Sopryazhennye uravneniya i algoritmy vozmuschenii, AN SSSR. Otdel vychislitelnoi matematiki, M., 1986 | MR