Multigrid in H(div) on Axisymmetric Domains

TL;DR

This paper develops and analyzes a multigrid algorithm for weighted H(div)-problems on axisymmetric domains, demonstrating uniform convergence on convex domains using Fourier finite element spaces.

Contribution

It introduces a multigrid method tailored for axisymmetric H(div)-problems and proves its uniform convergence on convex domains with modern smoothers.

Findings

Multigrid V-cycle converges uniformly on convex axisymmetric domains.

Fourier finite element spaces effectively handle axisymmetric H(div)-problems.

The method is applicable to general data in weighted H(div)-problems.

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.