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Convergence rate of some domain decomposition methods for overlapping and nonoverlapping subdomains. (English) Zbl 0873.65108
The problem under consideration is the convection-diffusion equation $-\nu \Delta u + a\partial_x u + b\partial_y u + cu = f$ on the entire $$(x, y)$$-plane. Here, $$a$$, $$b$$, $$c$$, and $$\nu$$ are constants, with $$a$$ and $$\nu$$ positive and $$c$$ non-negative. The domain decomposition involves dividing the plane into a finite number of vertical strips which may or may not overlap. Rates of convergence are obtained for three iterative methods: (1) an additive Schwarz method (a Jacobi-type method), (2) alternate downwind-upwind Gauss-Seidel sweeps, and (3) successive downwind Gauss-Seidel sweeps. The proof is based on results from the algebra of formal languages.

##### MSC:
 65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs 65N06 Finite difference methods for boundary value problems involving PDEs 65F10 Iterative numerical methods for linear systems 35J25 Boundary value problems for second-order elliptic equations
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