Multigrid methods for 3$D$ $H(\mathbf{curl})$ problems with nonoverlapping domain decomposition smoothers

TL;DR

This paper develops V-cycle multigrid methods with novel nonoverlapping domain decomposition smoothers for 3D H(curl) problems, improving efficiency for vector field simulations.

Contribution

It introduces new smoothing techniques based on substructuring for H(curl) problems and provides convergence analysis and numerical validation.

Findings

New smoothers outperform scalar methods for H(curl) problems

Convergence analysis confirms effectiveness of the proposed multigrid approach

Numerical experiments demonstrate improved computational efficiency

Abstract

We propose V--cycle multigrid methods for vector field problems arising from the lowest order hexahedral N\'{e}d\'{e}lec finite element. Since the conventional scalar smoothing techniques do not work well for the problems, a new type of smoothing method is necessary. We introduce new smoothers based on substructuring with nonoverlapping domain decomposition methods. We provide the convergence analysis and numerical experiments that support our theory.