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Substructuring preconditioners for saddle-point problems arising from Maxwell’s equations in three dimensions. (English) Zbl 1048.65109
The authors construct and analyze fast solution methods for the algebraic saddle-point problems arising from the finite element discretization of the mixed variational formulation of the boundary value problem $\text{curl}(\alpha\text{ curl}(u))+ \gamma_0 \beta u= f$ and $\text{div}(\beta u)= g$ in the computational domain $\Omega$ with vanishing tangential component $u\times n$ of the vector-valued function $u$. The main result consists in the construction of an almost optimal substructuring (non-overlapping domain decomposition) preconditioner for the eventually regularized finite element matrix arising from the first equation above. Finally, using this result and a similar result for the Schur-complement preconditioner, one can solve the algebraic saddle saddle-point problems very efficiently by a Uzawa-like iteration.

65N30Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE)
65N55Multigrid methods; domain decomposition (BVP of PDE)
78M10Finite element methods (optics)
78A30Electro- and magnetostatics
35Q60PDEs in connection with optics and electromagnetic theory
65F10Iterative methods for linear systems
65F35Matrix norms, conditioning, scaling (numerical linear algebra)
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