Smoothers Based on Nonoverlapping Domain Decomposition Methods for $H(\mathbf{curl})$ Problems: A Numerical Study

TL;DR

This paper investigates multigrid algorithms with nonoverlapping domain decomposition smoothers for 3D $H(\mathbf{curl})$ problems, demonstrating their robustness and efficiency through numerical experiments.

Contribution

It introduces a novel multigrid approach using nonoverlapping domain decomposition smoothers for $H(\mathbf{curl})$ problems discretized with Nédélec edge elements.

Findings

Algorithms are robust across various problem settings.

Numerical results show high efficiency and convergence.

Method outperforms traditional smoothers in tests.

Abstract

This paper presents a numerical study on multigrid algorithms of $V$-cycle type for problems posed in the Hilbert space $H(\mathbf{curl})$ in three dimensions. The multigrid methods are designed for discrete problems originated from the discretization using the hexahedral N\'{e}d\'{e}lec edge element of the lowest-order. Our suggested methods are associated with smoothers constructed by substructuring based on domain decomposition methods of nonoverlapping type. Numerical experiments to demonstrate the robustness and the effectiveness of the suggested algorithms are also provided.