Multigrid in H(div) on axisymmetric domains

Abstract

In this paper, we will construct and analyze a multigrid algorithm that can be applied to weighted H(div) problems on a two-dimensional domain. These problems arise after performing a dimension reduction to a three-dimensional axisymmetric H(div) problem. We will use recently developed Fourier finite element spaces that can be applied to axisymmetric H(div) problems with general data. We prove that if the axisymmetric domain is convex, then the multigrid V-cycle with modern smoothers will converge uniformly with respect to the meshsize.