## Factorization of the convection-diffusion operator and the Schwarz algorithm.(English)Zbl 0826.65102

A linear convection-diffusion equation of the type $${\mathcal L}u= \Delta u + v \cdot \nabla u - u = f$$ is considered on a two-dimensional strip-shaped domain. The operator $$\mathcal L$$ is factorized into two parabolic problems either exactly or approximately, the boundary conditions being fitted with a domain decomposition method. In the latter case, variable coefficients are admitted, too. Then the Schwarz algorithm is used as an alternative solver and the convergence rate is proved and tested on illustrative numerical examples for the domain decomposition both into strips and into rectangles.

### 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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