Local Fourier analysis of Balancing Domain Decomposition by Constraints algorithms

TL;DR

This paper adapts local Fourier analysis to study the eigenvalue distributions and condition numbers of two- and three-level BDDC algorithms, providing insights into their convergence behavior and performance improvements.

Contribution

It introduces a novel adaptation of local Fourier analysis for BDDC algorithms, enabling better understanding and optimization of their convergence properties.

Findings

LFA accurately predicts condition numbers for 2D Laplacian problems.

Combining BDDC with weighted diagonal scaling improves performance.

Optimized weights via LFA enhance the effectiveness of BDDC variants.

Abstract

Local Fourier analysis is a commonly used tool for the analysis of multigrid and other multilevel algorithms, providing both insight into observed convergence rates and predictive analysis of the performance of many algorithms. In this paper, for the first time, we adapt local Fourier analysis to examine variants of two- and three-level balancing domain decomposition by constraints (BDDC) algorithms, to better understand the eigenvalue distributions and condition number bounds on these preconditioned operators. This adaptation is based on a modified choice of basis for the space of Fourier harmonics that greatly simplifies the application of local Fourier analysis in this setting. The local Fourier analysis is validated by considering the two dimensional Laplacian and predicting the condition numbers of the preconditioned operators with different sizes of subdomains. Several variants are analyzed, showing the two- and three-level performance of the "lumped" variant can be greatly improved when used in multiplicative combination with a weighted diagonal scaling preconditioner, with weight optimized through the use of LFA.