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**Substructure preconditioners for elliptic saddle point problems.**
*(English)*
Zbl 0795.65072

Summary: Domain decomposition preconditioners for the linear systems arising from mixed finite element discretizations of second-order elliptic boundary value problems are proposed. The preconditioners are based on subproblems with either Neumann or Dirichlet boundary conditions on the interior boundary. The preconditioned systems have the same structure as the nonpreconditioned systems. In particular, we shall derive a preconditioned system with conditioning independent of the mesh parameter \(h\). The application of the minimum residual method to the preconditioned systems is also discussed.

### MSC:

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

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

65F10 | Iterative numerical methods for linear systems |

65F35 | Numerical computation of matrix norms, conditioning, scaling |

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

### Keywords:

elliptic saddle point problems; domain decomposition; preconditioners; mixed finite element; second-order elliptic boundary value problems; minimum residual method
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\textit{T. Rusten} and \textit{R. Winther}, Math. Comput. 60, No. 201, 23--48 (1993; Zbl 0795.65072)

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### References:

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