Geodesic
A variational-difference method for solving the Dirichlet’s problem for pseudodifferential elliptic equation of arbitrary order
Proceedings of the Yerevan State University. Physical and mathematical sciences, no. 1 (1990), pp. 26-32 
G. R. Pogosyan. A variational-difference method for solving the Dirichlet’s problem for pseudodifferential elliptic equation of arbitrary order. Proceedings of the Yerevan State University. Physical and mathematical sciences, no. 1 (1990), pp. 26-32. http://geodesic.mathdoc.fr/item/UZERU_1990_1_a3/
@article{UZERU_1990_1_a3,
     author = {G. R. Pogosyan},
     title = {A variational-difference method for solving the {Dirichlet{\textquoteright}s} problem for pseudodifferential elliptic equation of arbitrary order},
     journal = {Proceedings of the Yerevan State University. Physical and mathematical sciences},
     pages = {26--32},
     year = {1990},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/UZERU_1990_1_a3/}
}
TY  - JOUR
AU  - G. R. Pogosyan
TI  - A variational-difference method for solving the Dirichlet’s problem for pseudodifferential elliptic equation of arbitrary order
JO  - Proceedings of the Yerevan State University. Physical and mathematical sciences
PY  - 1990
SP  - 26
EP  - 32
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/UZERU_1990_1_a3/
LA  - ru
ID  - UZERU_1990_1_a3
ER  - 
%0 Journal Article
%A G. R. Pogosyan
%T A variational-difference method for solving the Dirichlet’s problem for pseudodifferential elliptic equation of arbitrary order
%J Proceedings of the Yerevan State University. Physical and mathematical sciences
%D 1990
%P 26-32
%N 1
%U http://geodesic.mathdoc.fr/item/UZERU_1990_1_a3/
%G ru
%F UZERU_1990_1_a3

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

In this paper we have obtained a variational-difference scheme, which solves the Dirichlet’s problem for $Au = f$, equations, where $A$ is a pseudodifferential operator according to a symbol $a(\xi)$, which satisfies the following condition: $c_1(1+|\xi|)^{\ast}\leq | a(\xi) \leq c_2(1+|\xi|)^{\ast}$. It has been proved that for the considered scheme the convergence speed order in the $\mathrm{H}_p(\Omega)$ space is equal to 1, and in the $L_2(\Omega)$ space it is $p+1$. The matrix of the obtained algebraic eqyation has a shape of a band with $2p+1$ width.

[1] G. R. Pogosyan, “Variatsionno-raznostnaya skhema resheniya zadachi Dirikhle dlya ellipticheskikh psevdodifferentsialnykh uravnenii vtorogo poryadka”, Uchenye zap. EGU, 1989, no. 1, 21–28

[2] M. I. Vishik, G. I. Eskin, “Ellipticheskie uravneniya v svertkakh v ogranichennoi oblasti i ikh prilozheniya”, Usp. mat. nauk, 22:1 (1965), 15–76 | DOI | MR | Zbl

[3] G. I. Marchuk, Yu. M. Agoshkov, Vvedenie v proektsionno-setochnye metody, Nauka, M., 1981 | MR | Zbl

[4] G. I. Marchuk, Metody vychislitelnoi matematiki, Nauka, M., 1980 | MR