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Multigrid in H(div) and H(curl). (English) Zbl 0974.65113
The authors are concerned with multigrid methods for 3D variational problems in the Hilbert spaces H(div) and H(curl) which naturally arise in problems of continuum mechanics and electromagnetism. The main result of the paper states that the standard V-cycle multigrid technique is both an efficient solver and preconditioner provided that appropriate finite dimensional subspaces of H(div) and H(curl) are chosen and suitable additive or multiplicative Schwarz smoothers are used. The analysis relies on discrete Helmholtz decompositions into subspaces of irrotational and solenoidal vector fields.

MSC:
65N55Multigrid methods; domain decomposition (BVP of PDE)
35J05Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation
65N30Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE)