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**Numerical experiments on a domain decomposition algorithm for nonlinear elliptic boundary value problems.**
*(English)*
Zbl 0656.65097

We present numerical experiments on a domain decomposition algorithm for nonlinear elliptic boundary value problems in planar domains. There has recently been much progress in the development of such algorithms for linear elliptic problems. These have focused on variational characterizations of the problem and on the preconditioning of the Schur complement associated with the decomposition. Although these could be used as part of a global Newton-type iterative scheme to solve a nonlinear problem, we choose the alternate path of first decomposing the problem and then applying an iterative method.

Our motivation for this is two-fold; first, we expect it will lead to algorithms which will require less communication between subproblems, an attractive property for implementations on parallel processors, second, this approach has even been found to be more efficient for serial computations in some cases. The essential step in this method is the solution of what we call the basic equations, a nonlinear analogue of the Schur complement problem. We are particularly concerned with the choice of boundary conditions at boundaries where subdomains intersect and their effect on the basic equations.

Our motivation for this is two-fold; first, we expect it will lead to algorithms which will require less communication between subproblems, an attractive property for implementations on parallel processors, second, this approach has even been found to be more efficient for serial computations in some cases. The essential step in this method is the solution of what we call the basic equations, a nonlinear analogue of the Schur complement problem. We are particularly concerned with the choice of boundary conditions at boundaries where subdomains intersect and their effect on the basic equations.

### MSC:

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

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

65Y05 | Parallel numerical computation |

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

### Keywords:

domain decomposition algorithm; global Newton-type iterative scheme; parallel processors; Schur complement problem; choice of boundary conditions
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\textit{T. Hagstrom} et al., Appl. Math. Lett. 1, No. 3, 299--302 (1988; Zbl 0656.65097)

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### References:

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[3] | Bramble, J.H.; Pasciak, J.E.; Schatz, A.H., The construction of preconditioners for elliptic problems by substructuring: II, Math. comp., (1988), to appear. · Zbl 0623.65118 |

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[5] | Chan, T.; Resasco, D., A framework for the analysis and construction of domain decomposition preconditioners, (), 217-230, appearing in |

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