Applied Numerical Mathematics 84 (2014) 66–79 Contents lists available at ScienceDirect Applied Numerical Mathematics www.elsevier.com/locate/apnum Parallel spectral-element direction splitting method for incompressible Navier–Stokes equations ✩ Lizhen Chen a,c , Jie Shen a,b,∗ , Chuanju Xu a , Li-Shi Luo c a Fujian Provincial Key Laboratory on Mathematical Modeling and Scientific Computing and School of Mathematical Sciences, Xiamen University, 361005 Xiamen, China b Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA c Beijing Computational Science Research Center, 100084, Beijing, China a r t i c l e i n f o a b s t r a c t Article history: Received 25 February 2013 Received in revised form 4 October 2013 Accepted 1 May 2014 Available online 14 June 2014 Keywords: Navier–Stokes equations Direction splitting Spectral-element methods Parallel algorithm An efficient parallel algorithm for the time dependent incompressible Navier–Stokes equations is developed in this paper. The time discretization is based on a direction splitting method which only requires solving a sequence of one-dimensional Poisson type equations at each time step. Then, a spectral-element method is used to approximate these one-dimensional problems. A Schur-complement approach is used to decouple the computation of interface nodes from that of interior nodes, allowing an efficient parallel implementation. The unconditional stability of the full discretized scheme is rigorously proved for the two-dimensional case. Numerical results are presented to show that this algorithm retains the same order of accuracy as a usual spectral-element projection type schemes but it is much more efficient, particularly on massively parallel computers.  2014 IMACS. Published by Elsevier B.V. All rights reserved. 1. Introduction The alternating direction method was developed in the 1950s for solving elliptic and parabolic equations (cf. [14]). It received new attention recently due to its suitability for parallel computing. In particular, Guermond and Minev [3] (see also [7]) proposed a direction splitting method which combines the pressure-stabilization method (cf. [17,8]) and the alternating direction technique for the time discretization of the incompressible Navier–Stokes equations. This scheme leads to a sequence of one-dimensional problems at each time step, making it very efficient and amenable for parallel computing. In [5] and [6], the authors implemented this direction splitting scheme with a finite difference method in space, and showed that the algorithm can achieve a high level of parallelism. In [1], the authors developed a direction splitting scheme with a hybrid (one domain) spectral method in space, and proved that the scheme is unconditional stable. However, the (one domain) spectral method is not suitable for massive parallel computers and for more general domains, so its applicability to large scale simulations is quite limited. The main purpose of this paper is to develop a stable direction splitting scheme with a spectral-element discretization in space that allows for efficient parallel implementation and is applicable for more general domains. The construction of the stable full discretized scheme is by no means a trivial extension of the spectral method in [1]. In order to prove the unconditional stability, we have to construct a new spectral-element method for the velocity while using the classical spectral-element method for the pressure. ✩ This research is partially supported by National Natural Science Foundation of China (Grant numbers 11071203, 11371298 and 91130002). * Corresponding author at: Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA. http://dx.doi.org/10.1016/j.apnum.2014.05.010 0168-9274/ 2014 IMACS. Published by Elsevier B.V. All rights reserved.