Variational problems arising from the inner products in the spaces H(div,Omega) and H(curl,Omega) typically lack the kind of ellipticity that makes plain multigrid work in the case of second order elliptic problems. The fault lies with the large kernels of the corresponding differential operators. They contain numerous highly oscillatory eigenfunctions that, however, fail to be associated with large eigenvalues. Conventional local smoothing is doomed, thus.
A remedy is offered by Helmholtz-decompositions, orthogonal splittings of the spaces into the kernel of the differential operator and its orthogonal complement. It turns out that the smoother remains effective with respect to error components in the latter. The former can be tackled based on a representation through potentials.
What renders this idea computationally feasible is the existence of discrete potentials in a finite element setting, provided that the appropriate finite element schemes are used. Those are Raviart-Thomas elements for H(div) and Nédélec's elements for H(curl,Omega).
Incorporating smoothing in potential space yields a multigrid method whose performance matches the usual multigrid efficiency for 2nd order problems. This talk will discuss approaches to the theoretical analysis of the scheme.