On a local Fourier analysis for overlapping block smoothers on triangular grids

TL;DR

This paper develops a local Fourier analysis method for overlapping block smoothers on triangular grids, applicable to various discretizations, and validates it through numerical experiments on fluid and electromagnetic problems.

Contribution

It introduces a general local Fourier analysis framework for overlapping block smoothers on triangular grids, applicable to different discretizations and validated with numerical results.

Findings

Analysis accurately predicts convergence factors.

Numerical results confirm the effectiveness of the analysis.

Method applies to complex discretizations like Nédélec finite elements.

Abstract

A general local Fourier analysis for overlapping block smoothers on triangular grids is presented. This analysis is explained in a general form for its application to problems with different discretizations. This tool is demonstrated for two different problems: a stabilized linear finite element discretization of Stokes equations and an edge-based discretization of the curl-curl operator by lowest-order N\'ed\'elec finite element method. In this latter, special Fourier modes have to be considered in order to perform the analysis. Numerical results comparing two- and three-grid convergence factors predicted by the local Fourier analysis to real asymptotic convergence factors are presented to confirm the predictions of the analysis and show their usefulness.