A FETI-DP type domain decomposition algorithm for three-dimensional incompressible Stokes equations

TL;DR

This paper extends FETI-DP domain decomposition algorithms to 3D incompressible Stokes equations, providing a new analysis that simplifies the coarse space and proves condition number bounds independent of the number of subdomains.

Contribution

The paper introduces a new analysis for 3D FETI-DP algorithms that reduces coarse space complexity and establishes subdomain-independent condition number bounds.

Findings

Condition number bounds are independent of the number of subdomains.

The coarse velocity space can be simplified, reducing computational complexity.

Numerical experiments confirm the convergence of the algorithms in 2D and 3D.

Abstract

The FETI-DP algorithms, proposed by the authors in [SIAM J. Numer. Anal., 51 (2013), pp.~1235--1253] and [Internat. J. Numer. Methods Engrg., 94 (2013), pp.~128--149] for solving incompressible Stokes equations, are extended to three-dimensional problems. A new analysis of the condition number bound for using the Dirichlet preconditioner is given. An advantage of this new analysis is that the numerous coarse level velocity components, required in the previous analysis to enforce the divergence free subdomain boundary velocity conditions, are no longer needed. This greatly reduces the size of the coarse level problem in the algorithm, especially for three-dimensional problems. The coarse level velocity space can be chosen as simple as for solving scalar elliptic problems corresponding to each velocity component. Both Dirichlet and lumped preconditioners are analyzed using a same framework in this new analysis. Their condition number bounds are proved to be independent of the number of subdomains for fixed subdomain problem size. Numerical experiments in both two and three dimensions demonstrate the convergence rate of the algorithms.