## Multigrid Method for *H(div)* in Three Dimensions

Ralf Hiptmair

Lehrstuhl für Angewandte Mathematik I

Universität Augsburg

Universitätsstraße 14

D-86159 Augsburg

Germany

URL: http://wwwhoppe.math.uni-augsburg.de/~hiptmair

*Abstract*
Stable multilevel decompositions of H(div)-conforming
finite element spaces with respect to an inner product
(**j**,**v**)+*r*(div **j**,div **v**) (*r*>0)
form the core of multilevel approaches to the efficient iterative
solution of many problems, ranging from mixed discretizations
to first order least squares approaches.

For Raviart-Thomas finite elements in 2D Vassilevski and Wang
[2] suggested a multilevel splitting as the basis
for an optimal preconditioner. This talk examines the generalization
of their ideas to three dimensions, which is highly non-trivial, since
the representation of discrete solenoidal vector fields plays a key role:
In three dimensions we have to resort to H(curl) -conforming finite element
spaces
introduced by Nédélec [1] to treat divergence
free vector
fields. We end up with a degenerate variational problem for the
bilinear form (**curl**.,**curl**.), for which no
multilevel decompositions had been available.

The central result to be presented in the talk is that a plain
nodal BPX-type splitting of the nested Nédélec-spaces yields
a decomposition that is stable in the **curl**-seminorm
independently of the number of levels involved.
The proof first establishes an isomorphism between the
quotient spaces with respect to the kernel of
the **curl**-operator
and certain discrete spaces. Then a duality argument is used to
show the stability of the nodal multilevel decomposition. Some
regularity assumptions are required for this step. Finally, a discrete
extension procedure for Raviart-Thomas spaces finishes the proof
for the general setting.

Thus, from algebraic multilevel theory we infer the optimal efficiency
of both multilevel preconditioned PCG and multigrid methods.
Results from numerical experiments for model problems confirm the rapid
convergence
of the iterations.

**1**-
J. N´ED´ELEC,
*Mixed finite elements in
$R3$ *, Numer. Math.,
35 (1980), pp. 315-341.

**2**-
P. VASSILEVSKI AND J. WANG,
*Multilevel iterative methods for mixed
finite element discretizations of elliptic problems*, Numer. Math., 63
(1992), pp. 503-520.