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Unformatted text preview: Submitted to Comput. Methods Appl. Mech. Eng. AN OVERVIEW OF PROJECTION METHODS FOR INCOMPRESSIBLE FLOWS J.L. GUERMOND 1 , P. MINEV 2 , AND JIE SHEN 3 Abstract. We discuss in this paper a series of important numerical issues related to the analysis and implementation of fractional step methods, often referred to in the literature as projection methods, for incompressible flows. We classify a broad range of projection schemes into three classes, namely the pressurecorrection methods, the velocitycorrection methods, and the consistent splitting methods. For each class of schemes, we review the theoretical and numerical convergence results available in the literature as well as associated open questions. We summarize the essential results in a table which could serve as a reference to analysts and practitioners. Contents 1. Introduction 2 2. Notations and preliminaries 3 3. The pressurecorrection schemes 4 3.1. The nonincremental pressurecorrection scheme 4 3.2. The standard incremental pressurecorrection schemes 4 3.3. The rotational incremental pressurecorrection schemes 6 3.4. Generalization 6 3.5. Implementation 7 3.6. Relation with other schemes 8 3.7. Numerical tests 9 4. The velocitycorrection schemes 12 4.1. The nonincremental velocitycorrection scheme 12 4.2. The standard incremental velocitycorrection schemes 13 4.3. The rotational incremental velocitycorrection schemes 14 4.4. Implementation 14 4.5. Numerical experiments 15 4.6. Relation with other schemes 15 5. Consistent splitting schemes 16 5.1. The key idea 16 5.2. Standard splitting scheme 16 5.3. Consistent splitting scheme 17 5.4. Numerical experiments 18 5.5. Relation with the gauge method 19 6. Inexact factorization schemes 21 Date : Draft version: February 17, 2005. 1991 Mathematics Subject Classification. 65N35, 65N22, 65F05, 35J05. Key words and phrases. NavierStokes equations, Projection methods, fractional step methods, incompress ibility, Finite elements, Spectral approximations. 1 LIMSI (CNRSUPR 3152), BP 133, 91403, Orsay, France ( guermond@limsi.fr ). The work of this author has been supported by CNRS and ICES under a TICAM Visiting Faculty Fellowship. 2 Dept. of Math. & Stat. Sci., University of Alberta, Edmonton, Canada T6G 2G1 ( minev@ualberta.ca ). The work of this author is supported by a Discovery Grant of NSERC. 3 Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA. ( shen@math.purdue.edu ). The work of this author is partially supported by NFS grant DMS0311915. 1 2 J.L. GUERMOND, P. MINEV, AND JIE SHEN 6.1. The matrix setting 21 6.2. Inexact factorization enforces a Neumann B.C 22 6.3. Inexact factorization is one viewpoint among many others 23 6.4. Inexact factorization is as accurate as PDEprojection 24 7. Interpretation of convergence tests 24 7.1. On the importance of norms 25 7.2. A numerical illustration 26 8. Effect of the infsup condition 26 8.1. The naive point of view 26 8.2. The functional analysis point of view...
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