Taking the cue from stabilized Galerkin methods for scalar advection problems, we adaptthe technique to boundary value problems modeling the advection of magnetic fields. Weprovide rigorous a priori error estimates for both fully discontinuouspiecewise polynomial trial functions and -conforming finite elements.

We prove the discrete compactness property of the edge elements of any order on a classof anisotropically refined meshes on polyhedral domains. The meshes, made up oftetrahedra, have been introduced in [Th. Apel and S. Nicaise, Math. Meth. Appl.Sci. 21 (1998) 519–549]. They are appropriately graded nearsingular corners and edges of the polyhedron.

We consider the convergence theory of adaptive multigrid methods for second-order elliptic problems and Maxwell’s equations. The multigrid algorithm only performs pointwise Gauss-Seidel relaxations on new degrees of freedom and their “immediate” neighbors. In the context of lowest order conforming finite element approximations, we present a unified proof for the convergence of adaptive multigrid V-cycle algorithms. The theory applies to any hierarchical tetrahedral meshes with uniformly bounded shape-regularity measures. The convergence rates for both problems are uniform with respect to the number of mesh levels and the number of degrees of freedom. We demonstrate our convergence theory by two numerical experiments.