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Tools for simulating non-stationary incompressible flow via discretely divergence-free finite element models. (English) Zbl 0794.76051
We develop simulation tools for the non-stationary incompressible two- dimensional Navier-Stokes equations. The most important components of the finite element code are: the fractional step \(\vartheta\)-scheme, which is of second-order accuracy and strongly \(A\)-stable, for the time discretization; a fixed point defect correction method with adaptive step length control for the nonlinear problems (stationary Navier-Stokes equations); a modified upwind discretization of higher-order accuracy for the convective terms. Finally, the resulting nonsymmetric linear subproblems are treated by a special multigrid algorithm which is adapted to the quadrilateral non-conforming discretely divergence-free finite elements.

76M10 Finite element methods applied to problems in fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
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[1] , and , ’FEAT2D. Finite element analysis tools. User manual. Release 1\(\cdot\)3’, Tech. Rep., University of Heidelberg, 1992.
[2] and , Finite Element Methods for Navier-Stokes Equations, Springer, Berlin, 1986. · Zbl 0585.65077 · doi:10.1007/978-3-642-61623-5
[3] The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1976.
[4] ’Ein robustes und effizientes Mehrgitterverfahren zur Lösung der instationären, inkompressiblen 2-D Navier-Stokes-Gleichungen mit diskret divergenzfreien finiten Elementen’, Thesis, University of Heidelberg, 1991.
[5] Rannacher, Numer. Methods Partial Differ. Eqns 8 pp 97– (1992)
[6] Griffiths, Int. j. numer. methods fluids 1 pp 323– (1981)
[7] ’Multigrid techniques for a class of discretely divergence-free finite element spaces’, in preparation.
[8] Ohmori, RAIRO Numer. Anal. 18 pp 309– (1984)
[9] Tobiska, MMAN 23 pp 627– (1989)
[10] ’Full and weighted upwind finite element methods’, in and (eds), Splines in Numerical Analysis, Int. Seminar ISAM 89, Weissig, 1989, Akademie-Verlag, Berlin, 1989. · Zbl 0685.65074
[11] and , ’Non-newtonian flow prediction by divergence-free finite elements’, Tech. Rep. 687, SFB 123, University of Heidelberg, 1992.
[12] ’Numerical analysis of nonstationary fluid flow (a survey)’, Tech. Rep. 492, SFB 123, University of Heidelberg, 1988.
[13] and , ’Numerical methods for nonlinear problems in fluid dynamics’, Proc. Int. Seminar on Scientific Supercomputers, Paris, February 1987, North-Holland, Amsterdam, 198x.
[14] and , ’Artifical boundaries and flux and pressure conditions for the incompressible Navier-Stokes equations’, Tech. Rep. 681, SFB 123, University of Heidelberg, 1992.
[15] ’Visualization tools for the nonstationary Navier-Stokes equations’, Tech. Rep. 680, SFB 123, University of Heidelberg, 1992.
[16] Ghia, J. Comput. Phys. 48 pp 387– (1982)
[17] Schreiber, J. Comput. Phys. 49 pp 310– (1983)
[18] Vanka, J. Comput. Phys. 65 pp 138– (1985)
[19] Zhang, MMAN 24 pp 133– (1990)
[20] Morgan, Notes on Numerical Fluid Mechanics 9 (1984)
[21] An Album of Fluid Motion, Parabolic, Stanford, CA, 1982.
[22] ’A parallel multigrid algorithm for solving the Navier-Stokes equations on a transputer system’, Tech. Rep. 699, SFB 123, University of Heidelberg, 1992.
[23] Hecht, Anal. 15 pp 119– (1981)
[24] ’Numerical methods in singularly perturbed problems’, in H. G. Roos, A. Felgenhauer and L. Angermann (eds), Int. Seminar in Applied Mathematics, Int. Seminar ISAM 91, Dresden, 1991.
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