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Two-level additive Schwarz preconditioners for reduced integration methods

Filipe Cumaru, Alexander Heinlein, Joachim Schöberl

TL;DR

The paper addresses efficiently solving Stokes flow discretized with reduced integration that replaces incompressibility with a penalty term. It proposes a two-level overlapping additive Schwarz preconditioner using an RGDSW coarse space to ensure scalability in 3D domains. The authors detail the RGDSW coarse space construction, its interface-based basis, and the parallel implementation via FROSch and NGSolve, including domain decomposition and solver choices. Numerical experiments on a unit cube and a flow around a cylinder demonstrate good weak scalability and quantify how the penalty parameter ε affects iteration counts, showing a practical accuracy-performance trade-off.

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.

Two-level additive Schwarz preconditioners for reduced integration methods

TL;DR

The paper addresses efficiently solving Stokes flow discretized with reduced integration that replaces incompressibility with a penalty term. It proposes a two-level overlapping additive Schwarz preconditioner using an RGDSW coarse space to ensure scalability in 3D domains. The authors detail the RGDSW coarse space construction, its interface-based basis, and the parallel implementation via FROSch and NGSolve, including domain decomposition and solver choices. Numerical experiments on a unit cube and a flow around a cylinder demonstrate good weak scalability and quantify how the penalty parameter ε affects iteration counts, showing a practical accuracy-performance trade-off.

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.

Paper Structure

This paper contains 5 sections, 11 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Two-level Schwarz preconditioners for saddle point problems
  3. The RGDSW coarse space
  4. Parallel implementation
  5. Numerical results

Figures (3)

  • Figure 1: Interface of a non-overlapping domain decomposition in two (a) and three (b) dimensions. The vertex, edge and face interface components are marked in red, blue and green, respectively.
  • Figure 2: Velocity field solutions for the proposed examples. Left: flow in a unit cube (slice at $x = 0.5$); right: flow around a cylinder in a channel.
  • Figure 3: Numerical scalability results for the flow in a unit cube (left column) and around a cylinder (right column). Top row: number of iterations; middle row: total time to solve the system of equations (setup of the preconditioner and application of the PCG iterations); bottom row: time to set up the preconditioner (assembly and factorization of the local and coarse problems).