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**On the coupling of two dimensional hyperbolic and elliptic equations: Analytical and numerical approach.**
*(English)*
Zbl 0709.65092

Domain decomposition methods for partial differential equations, Proc. 3rd Int. Symp. Houston/TX (USA) 1989, 22-63 (1990).

[For the entire collection see Zbl 0695.00026.]

This paper analyzes a domain decomposition method for a linear singularly perturbed advection-diffusion equation. It is assumed that there are two subdomains, in one of which the diffusion terms may be ignored. The boundary condition used at the artificial interior interface depends on the direction of the characteristic curves. It is always required that the flux is continuous, but continuity of the solution is imposed only at the inflow portion of the subdomain in which the advection equation is solved. A numerical method involving iteration between the subdomains is introduced, and it is shown to converge, often in a finite number of steps. It is also shown that the solution may be obtained by a vanishing- viscosity argument.

This paper analyzes a domain decomposition method for a linear singularly perturbed advection-diffusion equation. It is assumed that there are two subdomains, in one of which the diffusion terms may be ignored. The boundary condition used at the artificial interior interface depends on the direction of the characteristic curves. It is always required that the flux is continuous, but continuity of the solution is imposed only at the inflow portion of the subdomain in which the advection equation is solved. A numerical method involving iteration between the subdomains is introduced, and it is shown to converge, often in a finite number of steps. It is also shown that the solution may be obtained by a vanishing- viscosity argument.

Reviewer: G.Hedstrom

### MSC:

65N55 | Multigrid methods; domain decomposition for boundary value problems involving PDEs |

65N22 | Numerical solution of discretized equations for boundary value problems involving PDEs |

65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |

35J25 | Boundary value problems for second-order elliptic equations |

35L20 | Initial-boundary value problems for second-order hyperbolic equations |

35B25 | Singular perturbations in context of PDEs |