A computational study of algebraic coarse spaces for two-level overlapping additive Schwarz preconditioners

TL;DR

This paper explores algebraic coarse spaces for two-level overlapping additive Schwarz preconditioners, introducing an algebraic multiscale finite element approach and comparing it with other energy-minimizing coarse spaces to improve scalability and efficiency.

Contribution

It introduces an algebraic formulation of MsFEM based on AMS for Schwarz methods and compares it with existing algebraic energy-minimizing coarse spaces.

Findings

AMS relates to other energy-minimizing coarse spaces.

Comparison shows advantages of AMS over GDSW and RGDSW.

Algebraic coarse spaces improve preconditioner robustness.

Abstract

The two-level overlapping additive Schwarz method offers a robust and scalable preconditioner for various linear systems resulting from elliptic problems. One of the key to these properties is the construction of the coarse space used to solve a global coupling problem, which traditionally requires information about the underlying discretization. An algebraic formulation of the coarse space reduces the complexity of its assembly. Furthermore, well-chosen coarse basis functions within this space can better represent changes in the problem's properties. Here we introduce an algebraic formulation of the multiscale finite element method (MsFEM) based on the algebraic multiscale solver (AMS) in the context of the two-level Schwarz method. We show how AMS is related to other energy-minimizing coarse spaces. Furthermore, we compare the AMS with other algebraic energy-minimizing spaces: the generalized Dryja-Smith-Widlund (GDSW), and the reduced dimension GDSW (RGDSW).