an:04155821 Zbl 0704.65090 Lions, P. L. On the Schwarz alternating method. III: A variant for nonoverlapping subdomains EN Domain decomposition methods for partial differential equations, Proc. 3rd Int. Symp. Houston/TX (USA) 1989, 202-223 (1990). 1990
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65N55 65N12 65N22 65F10 35J05 domain decomposition methods; Poisson equation; nonoverlapping subdomains; Schwartz alternating method; convergence; convection- diffusion problems [For the entire collection see Zbl 0695.00026.] This paper is the third part of a series of three articles by the author [part I: ibid., 1st Int. Symp., Paris 1987, 1-42 (1989; Zbl 0658.65090); Part II: ibid., 2nd Int. Symp., Los Angeles 1988, 47-70 (1989; Zbl 0681.65072)] on the Schwartz alternating method for solving partial differential equations. In this part, the iteration is based on a nonoverlapping decomposition of the domain $$\Omega =\Omega_ 1\cup...\cup \Omega_ m\cup \Sigma$$ into m subdomains $$\Omega_ i$$, where $$\Sigma =\cup \gamma_{ij}$$ and $$\gamma_{ij}=\partial \Omega_ i\cap \partial \Omega_ j\setminus \partial \Omega$$ with $$i\neq j.$$ For the model problem $$-\Delta u=f$$ in $$\Omega$$ and $$u=0$$ on $$\partial \Omega$$, the iteration sequence $$\{(u^ n_ i)_{i=1,...,m}\}_{n=0,1,...}$$ consists of m functions $$u^ n_ i(x)$$ defined on $${\bar \Omega}{}_ i$$, where, for given $$(u^ n_ j)_{j=1,...,m'}$$ $$u_ i^{n+1}$$ will be defined as the solution of the partial differential $$-\Delta u_ i^{n+1}=f$$ in $$\Omega_ i$$ under Fourier's boundary conditions $\partial u_ i^{n+1}/\partial n_{ij}+\lambda_{ij}u_ i^{n+1}=\partial u^ n_ j/\partial n_{ij}+\lambda_{ij}u^ n_ j\quad (\lambda_{ij}=\lambda_{ji}>0)$ on the interface $$\gamma_{ij}$$ (j$$\neq i)$$ and under the given boundary condition $$u_ i^{n+1}=0$$ on $$\partial \Omega_ i\cap \partial \Omega$$. Here $$n_{ij}$$ is the unit outward normal to $$\partial \Omega_ i$$ on $$\gamma_{ij}$$. The author shows weak convergence of $$u^ n_ i$$ to $$u|_{\Omega_ i}$$ in $$H^ 1(\Omega_ i)$$ and of $$u^ n_ i|_{\gamma_{ij}}$$ to $$u|_{\gamma_{ij}}$$ in $$H^{1/2}(\gamma_{ij})$$. Extensions to more complicated problems including convection-diffusion problems are given. U.Langer Zbl 0695.00026; Zbl 0658.65090; Zbl 0681.65072