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.

References

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.