The Schwarz alternation method in a subspace. (English. Russian original) Zbl 0611.35017

Matsokin, A. M.; Nepomnyashchikh, S. V.

Sov. Math. 29, No. 10, 78-84 (1985); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1985, No. 10(281), 61-66 (1985).

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.

Reviewer: I.Onciulescu

MSC:

35J25	Boundary value problems for second-order elliptic equations
46E35	Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35D05	Existence of generalized solutions of PDE (MSC2000)

Keywords:

iteration process; Sobolev space; subdomains; Schwarz alternation method