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An overlapping Schwarz method for spectral element solution of the incompressible Navier-Stokes equations. (English) Zbl 0904.76057
This paper considers problems encountered in large-scale spectral element simulations of unsteady incompressible flows. An accurate simulation of even two-dimensional flows can require hundreds of thousands of grid points if the Reynolds number is of the order of \(10^4\). The author has developed an additive overlapping Schwarz preconditioner for the computationally challenging pressure operator which arises when an Uzawa decoupling procedure is applied to the \(P_N- P_{N-2}\) spectral element formulation of the incompressible Navier-Stokes equations. The pressure preconditioner is derived from local finite element Laplacians based on a triangulation of the Gauss (pressure) points, coupled with a global coarse grid operator of the spectral element vertices. The Schwarz procedure yields significantly improved convergence rates over previously employed deflation/block-Jacobi-based schemes. The overall Navier-Stokes solution times for several production runs have been reduced by a factor of 5 with the development of this preconditioner.
Reviewer: J.Siekmann (Essen)

76M25 Other numerical methods (fluid mechanics) (MSC2010)
76D05 Navier-Stokes equations for incompressible viscous fluids
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
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[1] Barret, R.; Berry, M.; Chan, T.; Demmel, J.; Donato, J.; Dongarra, J.; Eijkhout, V.; Pozo, R.; Romine, C.; van der Vorst, H., Templates, (1994), SIAM Philadelphia
[2] Bernardi, C.; Maday, Y., A collocation method over staggered grids for the Stokes problem, Int. J. numer. methods fluids, 8, 537, (1988) · Zbl 0665.76037
[3] Perot, J.Blair, An analysis of the fractional step method, J. comput. phys., 108, 51, (1993) · Zbl 0778.76064
[4] Cai, X.-C., The use of pointwise interpolation in domain decomposition with non-nested meshes, SIAM J. sci. comput., 14, 250, (1995)
[5] M. Casarin, 1995, Quasi-optimal Schwarz methods for the conforming spectral element discretization, Department of Computer Science, Courant Institute, NYU · Zbl 0889.65123
[6] M. Casarin, 1996, Schwarz Preconditioners for Spectral and Mortar Finite Element Methods with Applications to Incompressible Fluids, Courant Institute of Math. Sci. NYU
[7] T. F. Chan, B. F. Smith, J. Zou, 1994, Overlapping Schwarz Methods on Unstructured Meshes Using Non-matching Coarse Grids, Department of Math. UCLA
[8] W. Couzy, 1995, Spectral Element Discretization of the Unsteady Navier-Stokes Equations and Its Iterative Solution on Parallel Computers, École Polytechnique Fédérale de Lausanne
[9] Damaret, P.; Deville, M.O., Chebyshev pseudo-spectral solution of the Stokes equations using finite element preconditioning, J. comput. phys., 83, 463, (1989) · Zbl 0672.76039
[10] Deville, M.O.; Mund, E.H., Finite element preconditioning for pseudospectral solutions of elliptic problems, SIAM J. statist. comput., 11, 311, (1990) · Zbl 0701.65075
[11] Deville, M.O.; Mund, E.H., Fourier analysis of finite element preconditioned collocation schemes, SIAM J. sci. statist. comput., 13, 596, (1992) · Zbl 0745.65060
[12] M. Dryja, O. B. Widlund, 1987, An Additive Variant of the Schwarz Alternating Method for the Case of Many Subregions, Department of Computer Science, Courant Institute, NYU
[13] M. Dryja, 1989, An additive Schwarz algorithm for two- and three-dimensional finite element elliptic problems, 3rd Int. Symp. on Domain Decomposition Methods, 168, SIAM, Philadelphia · Zbl 0681.65075
[14] P. F. Fischer, 1989, Spectral Element Solution of the Navier-Stokes Equations on High Performance Distributed-Memory Parallel Processors, Massachusetts Institute of Technology
[15] P. F. Fischer, 1993, Projection Techniques for Iterative Solution ofAx_b_
[16] Fischer, P.F., Parallel domain decomposition for incompressible fluid dynamics, Contemp. math., 157, 313, (1994) · Zbl 0796.76065
[17] Fischer, P.F.; Rønquist, E.M., Spectral element methods for large scale parallel navier – stokes calculations, Comput. methods appl. mech. eng., 116, 69, (1994) · Zbl 0826.76060
[18] P. F. Fischer, 1996, Parallel multi-level solvers for spectral element methods, Proceedings, Int. Conf. on Spectral and High-Order Methods ’95, Houston, TX, A. V. IlinL. R. Scott, 595, Houston Journal of Mathematics
[19] Ghaddar, N.K.; Korczak, K.; Mikic, B.B.; Patera, A.T., Numerical investigation of incompressible flow in grooved channels. part 1: stability and self-sustained oscillations, J. fluid mech., 163, 99, (1986)
[20] Girault, V.; Raviart, P.A., Finite element approximation of the navier – stokes equations, (1979), Springer Berlin · Zbl 0396.65070
[21] W. D. Gropp, 1992, Parallel computing and domain decomposition, Fifth Conference on Domain Decomposition Methods for Partial Differential Equations, T. F. Chanet al. 349, SIAM, Philadelphia · Zbl 0770.65085
[22] Golub, G.H.; Vann Loan, C.F., Matrix computations, (1983), John Hopkins Univ. Press Baltimore
[23] Keyes, D.E.; Saad, Y.; Truhlar, D.G., Domain-based parallelism and problem decomposition methods in computational science and engineering, (1995), SIAM Philadelphia · Zbl 0829.00009
[24] Koumoutsakos, P.; Leonard, A., High-resolution simulations of the flow around an impulsively started cylinder using vortex methods, J. fluid mech., 296, 1, (1995) · Zbl 0849.76061
[25] Maday, Y.; Patera, A.T., Spectral element methods for the navier – stokes equations, (), 71
[26] Maday, Y.; Patera, A.T.; Rønquist, E.M., An operator-integration-factor splitting method for time-dependent problems: application to incompressible fluid flow, J. sci. comput., 5, 310, (1990) · Zbl 0724.76070
[27] Y. Maday, A. T. Patera, E. M. Rønquist, ThePN−PNmethod for the approximation of the Stokes problem, Numer. Math.
[28] Malik, M.R.; Zang, T.A.; Hussaini, M.Y., A spectral collocation method for the navier – stokes equations, J. comput. phys., 61, 64, (1985) · Zbl 0573.76036
[29] Mansfield, L., On the use of deflation to improve the convergence of conjugate gradient iteration, Commun. appl. numer. methods, 4, 151, (1988) · Zbl 0638.65024
[30] S. V. Nepomnyaschikh, 1986, Domain Decomposition and Schwarz Methods in a Subspace for the Approximate Solution of Elliptic Boundary Value Problems, Computing Center of the Siberian Branch of the USSR Acad. of Sci. Novosibirsk, USSR
[31] Nicolaides, R.A., Deflation of conjugate gradients with applications to boundary value problems, SIAM J. numer. anal, 24, 355, (1987) · Zbl 0624.65028
[32] Orszag, S.A.; Kells, L.C., Transition to turbulence in plane Poiseuille flow and plane Couette flow, J. fluid mech., 96, 159, (1980) · Zbl 0418.76036
[33] Orszag, S.A., Spectral methods for problems in complex geometries, J. comput. phys., 37, 70, (1980) · Zbl 0476.65078
[34] S. S. Pahl, 1993, Schwarz Type Domain Decomposition Methods for Spectral Element Discretizations, Department of Comput. and Appl. Mathematics, Univ. of the Witwatersrand, Johannesburg, South Africa
[35] Parter, S.V.; Rothman, E.E., Preconditioning Legendre spectral collocation approximation to elliptic problems, SIAM J. numer. anal., 32, 333, (1995) · Zbl 0822.65093
[36] Patera, A.T., A spectral element method for fluid dynamics: laminar flow in a channel expansion, J. comput. phys., 54, 468, (1984) · Zbl 0535.76035
[37] Pavarino, L.F.; Widlund, O.B., A polylogarithmic bound for an iterative substructuring method for spectral elements in three dimensions, SIAM J. numer. anal., 33, (1996) · Zbl 0856.41007
[38] M. M. Rai, P. Moin, 1989, Direct simulations of turbulent flow using finite difference schemes, presented at the 27th Aerospace Sciences Meeting, Reno, NV, AIAA-89-0369 (1989)
[39] E. M. Rønquist, 1992, A domain decomposition method for elliptic boundary value problems: Application to unsteady incompressible fluid flow, Fifth Conference on Domain Decomposition Methods for Partial Differential Equations, T. F. Chanet al. 545, SIAM, Philadelphia · Zbl 0767.76056
[40] E. M. Rønquist, 1996, A domain decomposition solver for the steady Navier-Stokes equations, Proceedings, Int. Conf. on Spectral and High-Order Methods ’95, Houston, TX, A. V. IlinL. R. Scott, 469, Houston Journal of Mathematics
[41] Saad, Y., Iterative methods for sparse linear systems, (1996), PWS Publishing Boston · Zbl 1002.65042
[42] Smith, B.; Bjørstad, P.; Gropp, W., Domain decomposition, (1996), Cambridge Univ. Press Cambridge
[43] O. B. Widlund
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