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Nodal auxiliary space preconditioning in H(curl) and H(div) spaces. (English) Zbl 1153.78006
Summary: We develop and analyze a general approach to preconditioning linear systems of equations arising from conforming finite element discretizations of $\bold H(\bold{curl}, \Omega)$- and $\bold H(\bold{div}, \Omega)$-elliptic variational problems. The preconditioners exclusively rely on solvers for discrete Poisson problems. We prove mesh-independent effectivity of the preconditioners by using the abstract theory of auxiliary space preconditioning. The main tools are discrete analogues of so-called regular decomposition results in the function spaces $\bold H(\bold{curl}, \Omega)$ and $\bold H(\bold{div}, \Omega)$. Our preconditioner for $\bold H(\bold{curl}, \Omega)$ is similar to an algorithm proposed in [{\it R. Beck}, Algebraic multigrid by component splitting for edge elements on simplicial triangulations, Tech. rep. SC 99-40, ZIB, Berlin, Germany (1999)].

78M10Finite element methods (optics)
65N30Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE)
65N55Multigrid methods; domain decomposition (BVP of PDE)
65N22Solution of discretized equations (BVP of PDE)
65F10Iterative methods for linear systems
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