On the Schwarz alternating method. II: Stochastic interpretation and order properties.

*(English)*Zbl 0681.65072
Domain decomposition methods, Proc. 2nd Int. Symp., Los Angeles/Calif. 1988, 47-70 (1989).

[For the entire collection see Zbl 0675.00021.]

This paper is the second part of a series of three articles on the Schwarz alternating method (SAM) for solving partial differential equations. In part I [Domain decomposition methods for partial differential equations, 1st Int. Symp., Paris/France 1987, 1-42 (1988; Zbl 0658.65090)] the variational approach was applied to study the convergence properties of the SAM in the corresponding energy space \((H^ 1\) for second-order problems), whereas the classical maximum principle used in the second part leads to convergence estimates in the \(L_{\infty}\)-norm. Convergence rate estimates are given for two overlapping and weakly overlapping subdomains.

Special choices of the initial guess in the SAM yield sequences \(\{u^ n\}\) uniformly converging to the solution u of the considered boundary value problem from below \((u_ n\uparrow u)\) and from above \((u_ n\downarrow u)\). Generalizations to the decomposition into more than two subdomains, to degenerate equations and to parabolic problems are given. Finally, the author presents an interesting stochastic interpretation and the corresponding stochastic convergence analysis of the SAM.

This paper is the second part of a series of three articles on the Schwarz alternating method (SAM) for solving partial differential equations. In part I [Domain decomposition methods for partial differential equations, 1st Int. Symp., Paris/France 1987, 1-42 (1988; Zbl 0658.65090)] the variational approach was applied to study the convergence properties of the SAM in the corresponding energy space \((H^ 1\) for second-order problems), whereas the classical maximum principle used in the second part leads to convergence estimates in the \(L_{\infty}\)-norm. Convergence rate estimates are given for two overlapping and weakly overlapping subdomains.

Special choices of the initial guess in the SAM yield sequences \(\{u^ n\}\) uniformly converging to the solution u of the considered boundary value problem from below \((u_ n\uparrow u)\) and from above \((u_ n\downarrow u)\). Generalizations to the decomposition into more than two subdomains, to degenerate equations and to parabolic problems are given. Finally, the author presents an interesting stochastic interpretation and the corresponding stochastic convergence analysis of the SAM.

Reviewer: U.Langer

##### MSC:

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 |

65F10 | Iterative numerical methods for linear systems |

35J70 | Degenerate elliptic equations |

35K60 | Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations |

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

65H10 | Numerical computation of solutions to systems of equations |