This paper introduces a strategy for automatically generating a block preconditioner for solving the incompressible Navier-Stokes equations. We consider the "pressure convectiondiffusion preconditioners" proposed by Kay, Loghin, and Wathen [SIAM

Ray Tuminaro ttIn recent years, considerable effoR has been placed on developing efficient and robust solution algorithms for the incompressible Navier-Stokes equations based on preconditioned Krylov methods. These include physicsbased methods. such as SIMPLE. and ourelv aleebraic oreconditioners based on the aovroximation of the Schur . , u .A 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 paper we develop a taxonomy of these ideas based on an adaptation of a generalized approximate factorization of the Navier-Stokes system first presented in [25]. This taxonomv illuminates the similarities and differences amone these oreconditioners and the central role olaved bv -. , , 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 inflowloutflow systems on both structured and unstructured meshes. The numerical experiments are performed using MPSalsa, a stabilized finite element code.

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Abstract. We demonstrate the performance of a fast computational algorithm for modeling the design of a microfluidic mixing device. The device uses an electrokinetic process, induced charge electroosmosis [17], by which a flow through the device is driven by a set of charged obstacles in it. Its design is realized by manipulating the shape and orientation of the obstacles in order to maximize the amount of fluid mixing within the device. The computation entails the solution of a constrained optimization problem in which function evaluations require the numerical solution of a set of partial differential equations: a potential equation, the incompressible Navier-Stokes equations, and a mass transport equation. The most expensive component of the function evaluation (which must be performed at every step of an iteration for the optimization) is the solution of the Navier-Stokes equations. We show that by using some new robust algorithms for this task [11,16], based on certain preconditioners that take advantage of the structure of the linearized problem, this computation can be done efficiently. Using this computational strategy, in conjunction with a derivative-free pattern search algorithm for the optimization, applied to a finite element discretization of the problem, we are able to determine optimal configurations of microfluidic devices.1. Introduction. Improvements in techniques for manufacturing devices at small length scales have created a growing interest in the construction of miniature devices for use in biomedical screening and chemical analysis. These microfluidic devices manipulate fluid flows over small length scales, between 10 and 100 µm, with a low fluid volume, and correspondingly low Reynolds number. This results in laminar flow of the type commonly found in blood samples, bacterial cell suspensions, or protein/antibody solutions. Methods for controlling and manipulating fluids at such length scales are a key ingredient in this process [18]. However, robust strategies for pumping and mixing in microfluidic devices are in short supply. Although mixing is one of the most time-consuming steps in biological agent detection, research and development of microfluidic mixing systems is relatively new. In this paper, we develop an efficient numerical algorithm for modeling this process using Induced Charge Electro-osmosis (ICEO) [17]. Our goal is to use this model to determine an optimal mixing design for a microfluidic device by manipulating the shape of the obstructions in the flow domain.In the course of modeling the mixing process, we need to compute the numerical solution of a collection of partial differential equations: a potential equation, a mass transport equation, and the incompressible Navier-Stokes equations. Solving the third of these is by the far the most complex and time consuming and one of our aims is to demonstrate the utility of some new solution algorithms for performing this task efficiently. Moreover, the systems of equations have on the order 10 5 to 10 8 unknowns, and these set...