J. Non-Newtonian Fluid Mech. 166 (2011) 354–362 Contents lists available at ScienceDirect Journal of Non-Newtonian Fluid Mechanics journal homepage: www.elsevier.com/locate/jnnfm A fast and efficient iterative scheme for viscoelastic flow simulations with the DEVSS finite element method Wook Ryol Hwang a,c , Mark A. Walkley b , Oliver G. Harlen a,∗ a Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom b School of Computing, University of Leeds, Leeds LS2 9JT, United Kingdom c School of Mechanical and Aerospace Engineering, Research Center for Aircraft Parts Technology (ReCAPT), Gyeongsang National University, Jinju 660-701, South Korea article info Article history: Received 6 May 2010 Received in revised form 5 January 2011 Accepted 5 January 2011 Available online 20 January 2011 Keywords: Iterative solver Block preconditioning Algebraic multigrid method Viscoelastic flow simulation Discrete elastic-viscous stress splitting (DEVSS) abstract We present a new fast iterative solution technique for the large sparse-matrix system that is commonly encountered in the mixed finite-element formulation of transient viscoelastic flow simulation: the DEVSS (discrete elastic-viscous stress splitting) method. A block-structured preconditioner for the velocity, pres- sure and viscous polymer stress has been proposed, based on a block reduction of the discrete system, designed to maintain spectral equivalence with the discrete system. The algebraic multigrid method and the diagonally scaled conjugate gradient method are applied to the preconditioning sub-block sys- tems and a Krylov subspace iterative method (MINRES) is employed as an outer solver. We report the performance of the present solver through example problems in 2D and 3D, in comparison with the corresponding Stokes problems, and demonstrate that the outer iteration, as well as each block precon- ditioning sub-problem, can be solved within a fixed number of iterations. The required CPU time for the entire problem scales linearly with the number of degrees of freedom, indicating O(N) performance of this solution algorithm. © 2011 Elsevier B.V. All rights reserved. 1. Introduction In this work, we consider a fast and efficient iterative solu- tion technique for the numerical simulation of an incompressible isothermal viscoelastic fluid flow, for which the governing set of equations is given as follows: ∇ ·  = 0, in , (1) ∇ · u = 0, in , (2)  =-pI + 2 s D +  p , in . (3) Eqs. (1)–(3) are, respectively, equations for the momentum bal- ance, the continuity, the constitutive relation in the computational domain . The symbols , u, p, D, I,  p and  s are the stress, the velocity, the pressure, the rate of deformation tensor, the identity tensor, the polymer contribution to the extra stress tensor and the viscosity of a Newtonian solvent (or the effective Newtonian viscos- ity in a polymer melt with fast relaxation modes), respectively. To complete the problem, we need to specify the evolution equation for the extra polymer stress  p . For example, an Oldroyd-B fluid in ∗ Corresponding author. Tel: +44 113 343 5189; fax: +44 113 343 5090. E-mail addresses: wrhwang@gnu.ac.kr (W.R. Hwang), m.a.walkley@leeds.ac.uk (M.A. Walkley), o.g.harlen@leeds.ac.uk, oliver@maths.leeds.ac.uk (O.G. Harlen). a differential form is expressed as (with Eq. (3))   ∂ p ∂t + u · ∇ p - (∇u) T ·  p -  p · ∇u  +  p - 2 p D = 0, in , (4) where  is a characteristic relaxation time in the polymer system and  p is the polymer viscosity. The discrete elastic-viscous stress splitting (DEVSS) formulation, originally developed by Guénette and Fortin [1], is one of the stan- dard finite-element formulations for the viscoelastic flow problem described in Eqs. (1)–(4), as it enhances ellipticity of the momen- tum equation, even for  s = 0 and it allows further possibilities in the selection of the discretization spaces. (A thorough review on this topic has been presented by Baaijens [2].) We remark that Eq. (4), the evolution equation of the polymeric stress, is hardly ever solved implicitly, since this equation is easily separated from the remain- ing equations (Eqs. (1)–(3)) with a proper choice of time-stepping method for transient flow simulations: e.g. the Adams–Bashforth method [3] or an operator splitting method [4]. In this regard, the present study focuses on the decoupled solution of Eqs. (1)–(3), which could be viewed as an augmented Stokes problem, consider- ing  p as a known variable. In the DEVSS formulation, we introduce an extra variable e, the viscous polymer stress, e = 2 p D. (5) 0377-0257/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.jnnfm.2011.01.003