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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 $\text{curl}\left(\alpha \phantom{\rule{0.166667em}{0ex}}\text{curl}\left(u\right)\right)+{\gamma }_{0}\beta u=f$ and $\text{div}\left(\beta u\right)=g$ in the computational domain ${\Omega }$ with vanishing tangential component $u×n$ 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.
##### MSC:
 65N30 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE) 65N55 Multigrid methods; domain decomposition (BVP of PDE) 78M10 Finite element methods (optics) 78A30 Electro- and magnetostatics 35Q60 PDEs in connection with optics and electromagnetic theory 65F10 Iterative methods for linear systems 65F35 Matrix norms, conditioning, scaling (numerical linear algebra)