Block Preconditioners for the Navier-Stokes Equations

Abstract

In recent years, considerable effort has been placed on developing efficient and robust solution algorithms for the incompressible Navier--Stokes equations based on preconditioned Krylov methods. These include physics-based methods, such as SIMPLE, and purely algebraic preconditioners based on the approximation of the Schur complement. All these techniques can be represented as approximate block factorization (ABF) type preconditioners. The goal is to decompose the application of the preconditioner into simplified sub-systems in which scalable multi-level type solvers can be applied. In this dissertation we develop a taxonomy of these ideas based on an adaptation of a generalized approximate factorization of the Navier-Stokes system first presented in [45]. This taxonomy illuminates the similarities and differences among these preconditioners and the central role played by efficient approximation of certain Schur complement operators. We then present a parallel computational study that examines the performance of these methods and compares them to an additive Schwarz domain decomposition (DD) algorithm. Results are presented for two and three-dimensional steady state problems for enclosed domains and inflow/outflow systems on both structured and unstructured meshes. The numerical experiments are performed using MPSalsa, a stabilized finite element code. We have also tested the utility of these methods in a more realistic fluid setting by solving an optimization problem related to the shape and topology of a microfluidic mixing device. This flow is modeled by Induced Charged Electro-osmosis (ICEO) described in [54]. The numerical results are performed using Sundance, a tool for the development of finite-element solutions of partial differential equations.