A Parallel PSPG Finite Element Method for Direct Simulation of Incompressible Flow J¨ org Stiller 1 , Karel Fraˇ na 1 , Roger Grundmann 1 , Uwe Fladrich 2 , and Wolfgang E. Nagel 2 1 Institute for Aerospace Engineering, TU Dresden, D-01062 Dresden, Germany {stiller,frana,grundmann}@tfd.mw.tu-dresden.de 2 Center for High Performance Computing, TU Dresden, D-01062 Dresden, Germany {fladrich,nagel}@zhr.tu-dresden.de Abstract. We describe a consistent splitting approach to the pressure- stabilized Petrov-Galerkin finite element method for incompressible flow. The splitting leads to (almost) explicit predictor and corrector steps linked by an implicit pressure equation which can be solved very ef- ficiently. The overall second-order convergence is proved in numerical experiments. Furthermore, the parallel implementation of the method is discussed and its scalability for up to 120 processors of a SGI Origin 3800 system is demonstrated. A significant superlinear speedup is observed and can be attributed to cache effects. First applications to large-scale fluid dynamic problems are reported. 1 Introduction We are interested in direct numerical simulations (DNS) of transitional and tur- bulent flows. Traditionally, specialized finite difference or spectral methods are used for this purpose. Though very efficient, these methods are often restricted to simple configurations. Unstructured finite volume methods and finite element methods are more flexible and offer the potential benefit of easier incorporating adaptive techniques. On the other hand, they are computationally less efficient and more difficult to parallelize. Also, the discretization scheme has to be care- fully designed to meet the accuracy requirements for DNS. In this paper, we consider a pressure-stabilized Petrov/Galerkin finite ele- ment method (PSPG-FEM) based on linear shape functions [1]. In Section 2, we describe a splitting approach that is similar to common projection and fractional step methods (see, e.g. [2]) but novel in the context of PSPG-FEM. The splitting yields an implicit Poisson-type equation for the pressure and an almost explicit predictor-corrector scheme for the velocity. In Section 3, we discuss the imple- mentation on top of our in-house MG grid library [3]. Numerical accuracy and scalability of the method are examined in Section 4. In Section 5, we briefly dis- cuss the application to DNS of electromagnetic stirring with rotating magnetic fields. M. Danelutto, D. Laforenza, M. Vanneschi (Eds.): Euro-Par 2004, LNCS 3149, pp. 726–733, 2004. c  Springer-Verlag Berlin Heidelberg 2004