Two-level additive Schwarz preconditioners for reduced integration methods

TL;DR

This paper develops and analyzes a two-level additive Schwarz preconditioner tailored for reduced integration discretizations of the Stokes equations, enhancing scalability and parallel efficiency in solving large, ill-conditioned linear systems.

Contribution

It introduces a novel two-level additive Schwarz preconditioner with a RGDSW coarse space for reduced integration methods applied to Stokes problems, and discusses its parallel implementation.

Findings

Demonstrates improved numerical scalability of the preconditioner.

Shows effective parallel implementation using FROSch and NGSolve.

Validates the approach on fluid flow problems with large, ill-conditioned systems.

Abstract

Incompressible fluid flow problems appear frequently in different applications. The discretization of such problems may result in large and ill-conditioned systems of linear equations. We consider the case of the Stokes equations discretized using a reduced integration method which approximates the incompressibility constraint by a penalty term thus allowing the problem to be solved only in terms of the velocity unknowns. We investigate the numerical scalability of a two-level overlapping additive Schwarz method with a reduced dimension generalized Dryja-Smith-Widlund (RGDSW) coarse space. In addition, we discuss the parallel implementation of the examples using the Fast and Robust Overlapping Schwarz (FROSch) package for additive Schwarz preconditioners and the NGSolve library, which implements multiple finite element space formulations.