##
**On the Schwarz alternating method. I.**
*(English)*
Zbl 0658.65090

Domain decomposition methods for partial differential equations, 1st Int. Symp., Paris/France 1987, 1-42 (1988).

[For the entire collection see Zbl 0649.00019.]

The paper is the first part of a series of three articles of the author on the Schwarz alternating method for solving partial differential equations. In the present first part, the author gives a variational interpretation of this method. The further two parts will be concerned with the maximum principle as a basic tool for the convergence analysis and with various variants of the method.

In the present paper, the author starts from an interpretation of the Schwarz method as an iterated projection on closed subspaces connected with the domain decomposition method. This representation of the iterates immediately yields convergence and even convergence rate estimates. Then some quite interesting ideas for the parallelization of the Schwarz method are discussed. Finally, extensions of the method to nonlinear monotone problems, variational inequalities and evolution equations (parabolic and hyperbolic initial boundary value problems) are made.

The paper is the first part of a series of three articles of the author on the Schwarz alternating method for solving partial differential equations. In the present first part, the author gives a variational interpretation of this method. The further two parts will be concerned with the maximum principle as a basic tool for the convergence analysis and with various variants of the method.

In the present paper, the author starts from an interpretation of the Schwarz method as an iterated projection on closed subspaces connected with the domain decomposition method. This representation of the iterates immediately yields convergence and even convergence rate estimates. Then some quite interesting ideas for the parallelization of the Schwarz method are discussed. Finally, extensions of the method to nonlinear monotone problems, variational inequalities and evolution equations (parabolic and hyperbolic initial boundary value problems) are made.

Reviewer: U.Langer

### MSC:

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

65K10 | Numerical optimization and variational techniques |

65F10 | Iterative numerical methods for linear systems |

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

35J65 | Nonlinear boundary value problems for linear elliptic equations |

49J40 | Variational inequalities |

35G10 | Initial value problems for linear higher-order PDEs |

35K25 | Higher-order parabolic equations |

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

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

65Y05 | Parallel numerical computation |

65M99 | Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems |

65N99 | Numerical methods for partial differential equations, boundary value problems |