Geodesic
Multigrid method for \(H\text{(div)}\) in three dimensions
Electronic transactions on numerical analysis, Tome 6 (1997), pp. 133-152
We are concerned with the design and analysis of a multigrid algorithm for (div; $\Omega $)-elliptic H linear variational problems. The discretization is based on (div; $\Omega $)-conforming Raviart-Thomas elements. A H thorough examination of the relevant bilinear form reveals that a separate treatment of vector fields in the kernel of the divergence operator and its complement is paramount. We exploit the representation of discrete solenoidal vector fields as curls of finite element functions in so-called N$\acute $ed$\acute $elec spaces. It turns out that a combined nodal multilevel decomposition of both the Raviart-Thomas and N$\acute $ed$\acute $elec finite element spaces provides the foundation for a viable multigrid method. Its Gauß-Seidel smoother involves an extra stage where solenoidal error components are tackled.
Classification : 65N55, 65N30
Keywords: multigrid, raviart-Thomas finite elements, N$\acute $ed$\acute $elec's finite elements, multilevel, mixed finite elements 
@article{ETNA_1997__6__a10,
     author = {Hiptmair,  R.},
     title = {Multigrid method for {\(H\text{(div)}\)} in three dimensions},
     journal = {Electronic transactions on numerical analysis},
     pages = {133--152},
     year = {1997},
     volume = {6},
     zbl = {0897.65046},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ETNA_1997__6__a10/}
}
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%A Hiptmair,  R.
%T Multigrid method for \(H\text{(div)}\) in three dimensions
%J Electronic transactions on numerical analysis
%D 1997
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Hiptmair,  R. Multigrid method for \(H\text{(div)}\) in three dimensions. Electronic transactions on numerical analysis, Tome 6 (1997), pp. 133-152. http://geodesic.mathdoc.fr/item/ETNA_1997__6__a10/