INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS Int. J. Numer. Meth. Fluids (2012) Published online in Wiley Online Library (wileyonlinelibrary.com/journal/nmf). DOI: 10.1002/fld.3733 Efficiency and scalability of a two-level Schwarz algorithm for i

This paper studies the application of two-level Schwarz algorithms to several models of Computational Fluid Dynamics. The purpose is to build an algorithm suitable for elliptic and convective models. The sub-domain approximated solution relies on the incomplete lower-upper factorisation (ILU). The algebraic coupling between the coarse grid and the Schwarz preconditioner is discussed. The Deflation Method (DM) and the Balancing Domain Decomposition (BDD) Method are studied for introducing the coarse grid correction as a preconditioner. Standard coarse grids are built with the characteristic or indicator functions of the subdomains. The building of a set of smooth basis functions (analogous to smoothed-aggregation methods) is considered. A first test problem is the Poisson problem with a discontinuous coefficient. The two options are compared for the standpoint of coarse-grid consistency and for the gain in scalability of the global Schwarz iteration. The advection-diffusion model is then considered as a second test problem. Extensions to compressible flows (together with incompressible flows for comparison) are then proposed. Parallel applications are presented and their performance measured. Copyright c © 2011 John Wiley & Sons, Ltd.

[1]  Dimitri J. Mavriplis,et al.  High-order discontinuous Galerkin methods using an hp-multigrid approach , 2006, J. Comput. Phys..

[2]  Alain Dervieux,et al.  Unstructured volume‐agglomeration MG: Solution of the Poisson equation , 1994 .

[3]  Cornelis Vuik,et al.  A Comparison of Deflation and the Balancing Preconditioner , 2005, SIAM J. Sci. Comput..

[4]  Charbel Farhat,et al.  A Minimum Overlap Restricted Additive Schwarz Preconditioner and Applications in 3D Flow Simulations , 1998 .

[5]  David A. Patterson,et al.  Computer Architecture - A Quantitative Approach, 5th Edition , 1996 .

[6]  M. Paraschivoiu,et al.  Parallel Two-Level Methods for Three-Dimensional Transonic Compressible Flow Simulations on Unstructured Meshes* , 2000 .

[7]  Marzio Sala.  An algebraic 2‐level domain decomposition preconditioner with applications to the compressible Euler equations , 2002 .

[8]  William Gropp,et al.  Newton-Krylov-Schwarz Methods in CFD , 1994 .

[9]  Frédéric Guyomarc'h,et al.  A Deflated Version of the Conjugate Gradient Algorithm , 1999, SIAM J. Sci. Comput..

[10]  Hua Xiang,et al.  A two level domain decomposition preconditioner based on local Dirichlet-to-Neumann maps , 2010 .

[11]  David E. Keyes,et al.  Application of Newton-Krylov methodology to a three-dimensional unstructured Euler code , 1995 .

[12]  Patrick Le Tallec,et al.  Balancing Domain Decomposition for Plates , 1993 .

[13]  Marcus Sarkis,et al.  A scaled and minimum overlap restricted additive Schwarz method with application to aerodynamics , 2000 .

[14]  D. Keyes.  How Scalable is Domain Decomposition in Practice , 1998 .

[15]  Rainald Löhner,et al.  Deflated preconditioned conjugate gradient solvers for the Pressure-Poisson equation , 2008, J. Comput. Phys..

[16]  Patrick R. Amestoy,et al.  Multifrontal parallel distributed symmetric and unsymmetric solvers , 2000 .

[17]  Hua Xiang,et al.  A Coarse Space Construction Based on Local Dirichlet-to-Neumann Maps , 2011, SIAM J. Sci. Comput..

[18]  Alain Dervieux,et al.  Multilevel functional preconditioning for shape optimisation , 2006 .

[19]  Petr Vanek,et al.  Analysis of an algebraic Petrov--Galerkin smoothed aggregation multigrid method , 2008 .

[20]  J. Mandel.  Balancing domain decomposition , 1993 .

[21]  David A. Patterson,et al.  Computer Architecture: A Quantitative Approach , 1969 .

[22]  Susanne C. Brenner,et al.  Two-level additive Schwarz preconditioners for plate elements , 1996, Wuhan University Journal of Natural Sciences.

[23]  R. Nicolaides.  Deflation of conjugate gradients with applications to boundary value problems , 1987 .

[24]  Cornelis Vuik,et al.  A Comparison of Two-Level Preconditioners Based on Multigrid and Deflation , 2010, SIAM J. Matrix Anal. Appl..

[25]  Alain Dervieux,et al.  An Additive Multilevel Optimization Method and Its Application to Unstructured Meshes , 1997 .

[26]  Charbel Farhat,et al.  A Variational Multiscale Method for the Large Eddy Simulation of Compressible Turbulent Flows on Unstructured Meshes - Application to vortex shedding , 2004 .

[27]  Peter Gloor,et al.  Extensions and Improvements , 1997 .

[28]  Stéphane Lanteri,et al.  Parallel linear multigrid algorithms for the acceleration of compressible flow calculations , 2000 .

[29]  Guillaume Houzeaux,et al.  Deflated preconditioned conjugate gradient solvers for the pressure‐Poisson equation: Extensions and improvements , 2011 .

[30]  Alain Dervieux,et al.  A low-diffusion MUSCL scheme for LES on unstructured grids , 2004 .

[31]  Daniel B. Szyld,et al.  An Algebraic Convergence Theory for Restricted Additive Schwarz Methods Using Weighted Max Norms , 2001, SIAM J. Numer. Anal..

[32]  Alain Dervieux,et al.  Parallel simulation of three-dimensional complex flows: Application to two-phase compressible flows and turbulent wakes , 2007, Adv. Eng. Softw..

[33]  Henk A. van der Vorst,et al.  Bi-CGSTAB: A Fast and Smoothly Converging Variant of Bi-CG for the Solution of Nonsymmetric Linear Systems , 1992, SIAM J. Sci. Comput..

[34]  J. Pasciak,et al.  Parallel multilevel preconditioners , 1990 .

[35]  Xiao-Chuan Cai,et al.  A Restricted Additive Schwarz Preconditioner for General Sparse Linear Systems , 1999, SIAM J. Sci. Comput..