an:03987790
Zbl 0611.35017
Matsokin, A. M.; Nepomnyashchikh, S. V.
The Schwarz alternation method in a subspace
EN
Sov. Math. 29, No. 10, 78-84 (1985); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1985, No. 10(281), 61-66 (1985).
00168259
1985
j
35J25 46E35 35D05
iteration process; Sobolev space; subdomains; Schwarz alternation method
On the domain \(\Omega \subset R^ 2\) with the Lipschitz boundary \(\partial \Omega =\Gamma_ 0\cup \Gamma_ 1\), a problem with conditions on \(\Gamma_ 0\) and \(\Gamma_ 1\) is considered. \(\Gamma_ 0\) is a finite union of some curvilinear segments.
The problem is to be solved in the space
\[
W^ 1_ 2(\Omega,\Gamma_ 0)=\{v\in W_ 2'(\Omega)| \quad v(x)=0,\quad x\in \Gamma_ 0\},
\]
considering the bilinear form
\[
a(u,v)=\int_{\Omega}(\sum^{3}_{i,j=1}a_{ij}(x)(\partial u/\partial x_ j)(\partial v/\partial x_ i)+a_ 0(x)uv) d\Omega +\int_{\Gamma_ 1}\sigma uv ds
\]
and the linear functional \(\ell (v)=\int_{\Omega}fv d\Omega.\)
The domain \(\Omega\) is decomposed in a union of more simple subdomains. For the obtained subdomains the Schwarz alternation method is used. Then the solution of the boundary problem is obtained by the intermediate of an auxiliary problem on the union of the boundaries of the conidered subdomains.
I.Onciulescu