Local Multigrid in H(curl)

TL;DR

This paper develops a convergence theory for a local multigrid correction scheme with hybrid smoothing applied to H(curl)-elliptic problems discretized by edge elements on adaptively refined tetrahedral meshes, showing uniform convergence rates.

Contribution

It introduces a novel convergence analysis for local multigrid methods with hybrid smoothing in the H(curl) setting, extending multigrid theory to edge element discretizations on adaptively refined meshes.

Findings

Convergence rate is uniform with respect to mesh refinement steps.

The analysis relies on local Helmholtz-type decompositions.

Results extend multigrid theory to H(curl) problems on complex meshes.

Abstract

We consider H(curl)-elliptic variational problems on bounded Lipschitz polyhedra and their finite element Galerkin discretization by means of lowest order edge elements. We assume that the underlying tetrahedral mesh has been created by successive local mesh refinement, either by local uniform refinement with hanging nodes or bisection refinement. In this setting we develop a convergence theory for the the so-called local multigrid correction scheme with hybrid smoothing. We establish that its convergence rate is uniform with respect to the number of refinement steps. The proof relies on corresponding results for local multigrid in a H1-context along with local discrete Helmholtz-type decompositions of the edge element space.