# zbMATH — the first resource for mathematics

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.

##### MSC:
 76M10 Finite element methods applied to problems in fluid mechanics 76D05 Navier-Stokes equations for incompressible viscous fluids
Full Text:
##### References:
 [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.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.