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An algebraic multigrid method for finite element discretizations with edge elements. (English) Zbl 1071.65170

Summary: This paper presents an algebraic multigrid method for the efficient solution of the linear system arising from a finite element discretization of variational problems in H 0 (curl,Ω). The finite element spaces are generated by Nédélec’s edge elements.

A coarsening technique is presented, which allows the construction of suitable coarse finite element spaces, corresponding transfer operators and appropriate smoothers. The prolongation operator is designed such that coarse grid kernel functions of the curl-operator are mapped to fine grid kernel functions. Furthermore, coarse grid kernel functions are ‘discrete’ gradients. The smoothers proposed by R. Hiptmair [SIAM J. Numer. Anal. 36, 204–225 (1998; Zbl 0922.65081)] and D. Arnold, R. Falk and R. Winther [Numer. Math. 85, 197–217 (2000; Zbl 0974.65113)] are directly used in the algebraic framework.

Numerical studies are presented for 3D problems to show the high efficiency of the proposed technique.


MSC:
65N55Multigrid methods; domain decomposition (BVP of PDE)
78M10Finite element methods (optics)
65Y20Complexity and performance of numerical algorithms