An Efficient Iterative Method for the Generalized Stokes Problem

Abstract

The generalized Stokes problem, which arises frequently in the simulation of timedependent Navier-Stokes equations for incompressible fluid flow, gives rise to symmetric linear systems of equations. These systems are indefinite due to a set of linear constraints on the velocity, causing difficulty for most preconditioners and iterative methods. This paper presents a novel method to obtain a preconditioned linear system from the original one which is then solved by an iterative method. This new method generates a basis for the velocity space and solves a reduced system which is symmetric and positive definite. Numerical experiments indicating superior convergence compared to existing methods are presented. A natural extension of this method to elliptic problems is also proposed, along with theoretical bounds on the rate of convergence, and results of experiments demonstrating robust and effective preconditioning.

Direct Numerical Simulations of Fluid–Solid Systems Using the Arbitrary Lagrangian–Eulerian Technique

“…For more efficient implementations, one may take the advantage of the structure of the system in (53), and design different preconditioners for different parts of the equations within the system. The design of more efficient and reliable preconditioners, especially for parallel computation, is still the topic of active research [49,60,63].…”

A Preconditioner for Generalized Saddle Point Problems

“…Besides specialized sparse direct solvers [16,17] we mention, among others, Uzawa-type schemes [11,21,24,27,62], block and approximate Schur complement preconditioners [4,15,20,22,41,45,46,48,51], splitting methods [18,30,31,49,57], indefinite preconditioning [23,35,39,43,48], iterative projection methods [5], iterative null space methods [1,32,54], and preconditioning methods based on approximate factorization of the coefficient matrix [25,50]. Several of these algorithms are based on some form of reduction to a smaller system, for example, by projecting the problem onto the null space of B, while others work with the original (augmented) matrix in (1.1).…”

A New GSOR Method for Generalised Saddle Point Problems

“…In recent years, many iterative methods have been introduced to solve the problem (1.1), including Uzawa-type schemes [14,20,23,25,33,34,51,53], iterative projection methods [3], block and approximate Schur complement preconditioners [17,19,22,35,40,42,43], iterative null space methods [1,26,48], splitting methods [4,7,[9][10][11][12][13]18,29,30,36,38,39,41,46,50], indefinite preconditioning [31,37], and preconditioning methods based on approximate factorisation of the coefficient matrix [6,8,28,44]. A classical approach to solve (1.1) is the successive overrelaxation (SOR) iteration method [49], which can involve relatively low computation per iterative step.…”