A short course on numerical simulation of viscous flow:Discretization, optimization and stability analysis

Rolf Rannacher

doi:10.3934/dcdss.2012.5.1147

Article Contents

2012, Volume 5, Issue 6: 1147-1194. Doi: 10.3934/dcdss.2012.5.1147

This issue Previous Article The thermo-mechanics of rate-type fluids Next Article A framework for the development of implicit solvers for incompressible flow problems

A short course on numerical simulation of viscous flow: Discretization, optimization and stability analysis

Rolf Rannacher^1,

1.
Institute of Applied Mathematics, University of Heidelberg, Im Neuenheimer Feld 293/294, D-69120 Heidelberg

Received: November 2011

Revised: March 2012

Published: December 2012

Abstract Related Papers Cited by

Abstract

This article contains part of the material of four introductory lectures given at the 12th school ``Mathematical Theory in Fluid Mechanics'', Spring 2011, at Kácov, Czech Republic, on ``Numerical simulation of viscous flow: discretization, optimization and stability analysis''. In the first lecture on ``Numerical computation of incompressible viscous flow'', we discuss the Galerkin finite element method for the discretization of the Navier-Stokes equations for modeling laminar flow. Particular emphasis is put on the aspects pressure stabilization and truncation to bounded domains. In the second lecture on ``Goal-oriented adaptivity'', we introduce the concept underlying the Dual Weighted Residual (DWR) method for goal-oriented residual-based adaptivity in solving the Navier-Stokes equations. This approach is presented for stationary as well as nonstationary situations. In the third lecture on ``Optimal flow control'', we discuss the use of the DWR method for adaptive discretization in flow control and model calibration. Finally, in the fourth lecture on ``Numerical stability analysis'', we consider the numerical stability analysis of stationary flows employing the concepts of linearized stability and pseudospectrum.

Keywords:

Mathematics Subject Classification: Primary: 65M60, 65N30, 76M10; Secondary: 35Q30, 76D05.

Citation:

Related Papers

Cited by

References

[1]	M. Ainsworth and J. T. Oden, A posteriori error estimation in finite element analysis, Comput. Methods Appl. Mech. Eng., 142 (1997), 1-88.doi: 10.1016/S0045-7825(96)01107-3.
[2]	W. Bangerth and R. Rannacher, "Adaptive Finite Element Methods for Differential Equations," Lectures in Mathematics, Birkhäuser, 2003.
[3]	R. Becker, An adaptive finite element method for the Stokes equations including control of the iteration error, in Numerical Mathematics and Advanced Applications, Proc. of ENUMATH 1997 (Eds. H.G. Bock et al.), pp. 609-620, World Scientific, London, 1998.
[4]	R. Becker, An optimal-control approach to a posteriori error estimation for finite element discretizations of the Navier-Stokes equations, East-West J. Numer. Math., 9 (2000), 257-274.
[5]	R. Becker, Mesh adaptation for stationary flow control, J. Math. Fluid Mech., 3 (2001), 317-341.doi: 10.1007/PL00000974.
[6]	R. Becker and M. Braack, A finite element pressure gradient stabilization for the Stokes equations based on local projection, Calcolo, 38 (2001), 363-379.doi: 10.1007/s10092-001-8180-4.
[7]	R. Becker, M. Braack, D. Meidner, R. Rannacher and B. Vexler, Adaptive finite element methods for PDE-constrained optimal control problems, in Reactive Flow, Diffusion and Transport (W. Jäger et al., eds), pp. 177-205, Springer, Heidelberg-Berlin, 2006.
[8]	R. Becker, M. Braack, R. Rannacher and Th. Richter, Mesh and model adaptivity for flow problems, in Reactive Flows, Diffusion and Transport (W. Jäger et al., eds), pp. 47-75, Springer, Heidelberg-Berlin, 2006.
[9]	R. Becker, V. Heuveline and R. Rannacher, An optimal control approach to adaptivity in computational fluid mechanics, Int. J. Numer. Meth. Fluids, 40 (2001), 105-120.doi: 10.1002/fld.269.
[10]	R. Becker, H. Kapp and R. Rannacher, Adaptive finite element methods for optimal control of partial differential equations: basic concepts, SIAM J. Optimization Control, 39 (2000), 113-132.doi: 10.1137/S0363012999351097.
[11]	R. Becker and R. Rannacher, Finite element solution of the incompressible Navier-Stokes equations on anisotropically refined meshes, Proc. Workshop Fast Solvers for Flow Problems, Kiel, 1994, Vieweg, 1995.
[12]	R. Becker and R. Rannacher, Weighted a posteriori error control in FE methods, Lecture at ENUMATH-95, Paris, Sept. 18-22, 1995, Proc. ENUMATH'97 (H. G. Bock et al., eds), pp. 621-637, World Scientific, Singapore, 1998.
[13]	R. Becker and R. Rannacher, A feed-back approach to error control in finite element methods: basic analysis and examples, East-West J. Numer. Math., 4 (1996), 237-264.
[14]	R. Becker and R. Rannacher, An optimal control approach to a posteriori error estimation in finite element methods, Acta Numer., 10 (2001), 1-102.doi: 10.1017/S0962492901000010.
[15]	R. Becker and B. Vexler, A posteriori error estimation for finite element discretization of parameter identification problems, Numer. Math., 96 (2004), 435-459.doi: 10.1007/s00211-003-0482-9.
[16]	R. Becker, C. Johnson and R. Rannacher, Adaptive error control for multigrid finite element methods, Computing, 55 (1995), 271-288.doi: 10.1007/BF02238483.
[17]	C. Bertsch and V. Heuveline, On multigrid methods for the eigenvalue computation of non-selfadjoint elliptic operators, East-West J. Numer. Math., 8 (2000), 275-297.
[18]	M. Besier and R. Rannacher, Goal-oriented space-time adaptivity in the finite element Galerkin method for the computation of nonstationary incompressible flow, Int. J. Numer. Meth. Fluids, 2011, to appear.
[19]	M. Besier and W. Wollner, On the pressure approximation in nonstationary incompressible flow simulations on dynamically varying spatial meshes, Int. J. Numer. Methods Fluids, doi:10.1002/fld.2625, 2011.
[20]	S. Bönisch, Th. Dunne and R. Rannacher, Lecture on numerical simulation of liquid-structure interaction, in Hemodynamical Flows: Aspects of Modeling, Analysis and Simulation (G.P. Galdi et al., eds), Birkhäuser, Basel, 2007.
[21]	M. Braack and A. Ern, A posteriori control of modeling errors and discretization errors, Multiscale Model. Simul., 1 (2003), 221-238.doi: 10.1137/S1540345902410482.
[22]	M. Braack, E. Burman, V. John and G. Lube, Stabilized finite element methods for the generalized Oseen equations, Comput. Methods Appl. Mech. Engrg., 196 (2007), 853-866.doi: 10.1016/j.cma.2006.07.011.
[23]	M. Braack and R. Rannacher, "Adaptive Finite Element Methods for Low-Mach-Number Flows with Chemical Reactions," Lecture Series 1999-03, 30th Computational Fluid Dynamics, (H. Deconinck, ed.), v. Karman Inst. Fluid Dynamics, Belgium, 1999.
[24]	M. Braack and Th. Richter, Solutions of 3D Navier-Stokes benchmark problems with adaptive finite elements, Computers and Fluids, 35 (2006), 372-392.doi: 10.1016/j.compfluid.2005.02.001.
[25]	J. H. Bramble and J. E. Osborn, Rate of convergence estimates for nonselfadjoint eigenvalue approximations, Math. Comp., 27 (1973), 525-545.doi: 10.1090/S0025-5718-1973-0366029-9.
[26]	S. C. Brenner and R. L. Scott, "The Mathematical Theory of Finite Element Methods," Springer, Berlin-Heidelberg-New York, 1994.
[27]	F. Brezzi and M. Fortin, "Mixed and Hybrid Finite Element Methods," Springer, Berlin-Heidelberg-New York, 1991.
[28]	G. F. Carey and J. T. Oden, "Finite Elements, Computational Aspects, Vol III," Prentice-Hall, 1984.
[29]	A. J. Chorin, Numerical solution of the Navier-Stokes equations, Math.Comp., 22 (1968), 745-762.doi: 10.1090/S0025-5718-1968-0242392-2.
[30]	P. G. Ciarlet, "The Finite Element Methods for Elliptic Problems," North-holland, Amsterdam, 1978.
[31]	Th. Dunne and R. Rannacher, Adaptive finite element approximation of fluid-structure interaction based on an Eulerian variational formulation, in Fluid-Structure Interaction: Modelling, Simulation, Optimisation (H.-J. Bungartz and M. Schäfer, eds), pp. 110-145 Springer, Heidelberg, 2006.
[32]	W. E and J. G. Liu, Projection method I: convergence and numerical boundary layers, SIAM J. Numer. Anal., 32 (1995), 1017-1057.doi: 10.1137/0732047.
[33]	K. Eriksson, D. Estep, P. Hansbo and C. Johnson, Introduction to adaptive methods for differential equations, Acta Numerica 1995 (A. Iserles, ed.), pp. 105-158, Cambridge University Press, 1995.
[34]	M. Feistauer, "Mathematical Methods in Fluid Dynamics," Longman Scientific&Technical, England, 1993.
[35]	J. H. Ferziger and M. Peric, "Computational Methods for Fluid Dynamics," Springer, Berlin-Heidelberg-New York, 1996.
[36]	L. P. Franca and S. L. Frey, Stabilized finite element methods: II The incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 99 (1992), 209-233.doi: 10.1016/0045-7825(92)90041-H.
[37]	S. Friedlander and D. Serre, "Handbuch for Mathematical Fluid Dynamics, Vol. 1/2," North-Holland, Amsterdam, 2002.
[38]	G. P. Galdi, "An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. 1: Linearized Steady Problems, Vol. 2: Nonlinear Steady Problems," Springer: Berlin-Heidelberg-New York, 1994.
[39]	D. Gerecht, R. Rannacher and W. Wollner, Computational aspects of pseudospectra in hydrodynamic stability analysis, J. Math. Fluid Mech., 2011, to appear.doi: 10.1007/s00021-011-0085-7.
[40]	M. B. Giles and E. Süli, Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality, in Acta Numerica 2002 (A. Iserles, ed.), pp. 145-236, Cambridge University Press, 2002.
[41]	V. Girault and P.-A. Raviart, "Finite Element Methods for the Navier-Stokes Equations," Springer, Heidelberg, 1986.
[42]	R. Glowinski, Finite element methods for incompressible viscous flow, in Handbook of Numerical Analysis Volume IX: Numerical Methods for Fluids (Part 3) (P. G. Ciarlet and J. L. Lions, eds), North-Holland: Amsterdam, 2003.
[43]	R. Glowinski, J. Hron, L. Rivkind and S. Turek, Numerical study of a modified time-stepping theta-scheme for incompressible flow simulations, J. Scientific Comput., 28 (2006), 533-547, doi: 10.1007/s10915-006-9083-y.
[44]	P. M. Gresho and R. L. Sani, "Incompressible Flow and the Finite Element Method," John Wiley & Sons, Chichester, 1998.
[45]	V. Heuveline and R. Rannacher, A posteriori error control for finite element approximations of elliptic eigenvalue problems, Adv. Comput. Math., 15 (2001), 1-32.doi: 10.1023/A:1014291224961.
[46]	V. Heuveline and R. Rannacher, Adaptive FE eigenvalue approximation with application to hydrodynamic stability analysis, in Advances in Numerical Mathematics (W. Fitzgibbon et al., eds), pp. 109-140, Inst. of Numer. Math. RAS, Moscow, 2006.
[47]	J. G. Heywood, The Navier-Stokes equations: On the existence, regularity and decay of solutions, Indiana Univ. Math. J., 29 (1980), 639-681.doi: 10.1512/iumj.1980.29.29048.
[48]	J. Heywood and R. Rannacher, Finite element approximation of the nonstationary Navier-Stokes problem, Part I: Regularity of solutions and second-order error estimates for spatial discretization, SIAM J. Numer. Anal., 19 (1982), 275-311.doi: 10.1137/0719018.
[49]	J. Heywood and R. Rannacher, Finite element approximation of the nonstationary Navier-Stokes problem, Part II: Stability of solutions and error estimates uniform in time, SIAM J. Numer. Anal., 23 (1986), 750-777.doi: 10.1137/0723049.
[50]	J. G. Heywood and R. Rannacher, An analysis of stability concepts for the Navier-Stokes equations, J. Reine Angew. Math., 372 (1986), 1-33.
[51]	J. G. Heywood and R. Rannacher, Finite element approximation of the nonstationary Navier-Stokes problem, Part III: Smoothing property and higher-order error estimates for spatial discretization, SIAM J. Numer. Anal., 25 (1988), 489-512.doi: 10.1137/0725032.
[52]	J. G. Heywood and R. Rannacher, Finite element approximation of the nonstationary Navier-Stokes problem, Part IV: Error analysis for second-order time discretization, SIAM J. Numer. Anal., 27 (1990), 353-384.doi: 10.1137/0727022.
[53]	J. G Heywood, R. Rannacher and S. Turek, Artificial boundaries and flux and pressure conditions for the incompressible Navier-Stokes equations, Int. J. Numer. Meth. Fluids, 22 (1996), 1-28.doi: 10.1002/(SICI)1097-0363(19960315)22:5<325::AID-FLD307>3.0.CO;2-Y.
[54]	T. J. R. Hughes and A. N. Brooks, Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equation, Comput. Meth. Appl. Mech. Engrg. 32 (1982), 199-259.doi: 10.1016/0045-7825(82)90071-8.
[55]	T. J. R. Hughes, L. P. Franca and M. Balestra, A new finite element formulation for computational fluid mechanics: V. Circumventing the Babuska-Brezzi condition: A stable Petrov-Galerkin formulation of the Stokes problem accommodating equal order interpolation, Comput. Meth. Appl. Mech. Engrg., 59 (1986), 85-99.doi: 10.1016/0045-7825(86)90025-3.
[56]	V. John, Higher order finite element methods and multigrid solvers in a benchmark problem for the 3D Navier-Stokes equations, Int. J. Numer. Meth. Fluids, 40 (2002), 775-798.doi: 10.1002/fld.377.
[57]	C. Johnson, S. Larsson, V. Thoméand L. B. Wahlbin, Error estimates for spatially discrete approximations of semilinear parabolic equations with nonsmooth initial data, Math. Comp., 49 (1987), 331-357.doi: 10.1090/S0025-5718-1987-0906175-1.
[58]	C. Johnson, R. Rannacher and M. Boman, Numerics and hydrodynamic stability: Towards error control in CFD, SIAM J. Numer. Anal., 32 (1995), 1058-1079.doi: 10.1137/0732048.
[59]	S. Kracmar and J. Neustupa, Modelling of flows of a viscous incompressible fluid through a channel by means of variational inequalities, Z. Angew. Math. Mech., 74 (1994), T637-T639.
[60]	O. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow," Gordon and Breach, London, 1969.
[61]	M. Luskin and R. Rannacher, On the smoothing property of the Galerkin method for parabolic equations, SIAM J. Numer. Anal., 19 (1981), 93-113.doi: 10.1137/0719003.
[62]	M. Luskin and R. Rannacher, On the smoothing property of the Crank-Nicolson scheme, Appl. Anal., 14 (1982), 117-135.doi: 10.1080/00036818208839415.
[63]	L. Machiels, A. T. Patera and J. Peraire, Output bound approximation for partial differential equations, application to the incompressible Navier-Stokes equations, in Industr. and Environm. Appl. of Direct and Large Eddy Numerical Simulation (S. Biringen, ed.), Springer, Berlin-Heidelberg-New York, 1998.
[64]	D. Meidner, "Adaptive Space-Time Finite Element Methods for Optimization Problems Governed by Nonlinear Parabolic Systems," doctoral thesis, University of Heidelberg, 2007.
[65]	D. Meidner, R. Rannacher and J. Vihharev, Goal-oriented error control of the iterative solution of finite element equations, J. Numer. Math., 17 (2009), 143-172.doi: 10.1515/JNUM.2009.009.
[66]	S. Müller-Urbaniak, "Eine Analyse des Zwischenschritt-$\theta$-Verfahrens zur Lösung der instationären Navier-Stokes-Gleichungen," doctoral thesis, Preprint 93-01, SFB 359, University of Heidelberg, 1993.
[67]	S. Müller, A. Prohl, R. Rannacher and S. Turek, Implicit time-discretization of the nonstationary incompressible Navier-Stokes equations, in Proc. GAMM Seminar Fast Solvers for Flow Problems, Kiel, (W.Hackbusch, G.Wittum, eds), pp. 175-191, NNFM, Vol.49, Vieweg, Braunschweig, 1994.
[68]	J. T. Oden, W. Wu and M. Ainsworth, An a posteriori error estimate for finite element approximations of the Navier-Stokes equations, Comput. Methods Appl. Mech. Eng., 111 (1993), 185-202.doi: 10.1016/0045-7825(94)90045-0.
[69]	O. Pironneau, "Finite Element Methods for Fluids," John Wiley, Chichester, 1983.
[70]	A. Prohl, "Projection and Quasi-Compressibility Methods for Solving the Incompressible Navier-Stokes Equations," Teubner, Stuttgart, 1997.
[71]	L. Quartapelle, "Numerical Solution of the Incompressible Navier-Stokes Equations," Birkhäuser, Basel, 1993.
[72]	R. Rannacher, On Chorin's projection method for the incompressible Navier-Stokes equations, in Navier-Stokes Equations: Theory and Numerical Methods (J. G. Heywood et al., eds), pp. 167-183, LNM, Vol. 1530, Springer, Berlin-Heidelberg-New York, 1992.
[73]	R. Rannacher, Finite methods for the incompressible Navier-Stokes equations, in Fundamental Directions in Mathematical Fluid Mechanics (G.P. Galdi et al., eds), pp. 191-216, Birkhäuser, Basel-Boston-Berlin, 2000.
[74]	R. Rannacher, Incompressible viscous flow, in Encyclopedia of Computational Mechanics (E. Stein et al., eds), Volume 3 "Fluids,'' John Wiley, Chichester, 2004.
[75]	R. Rannacher, Methods for numerical flow simulation, in Hemodynamical Flows: Aspects of Modeling, Analysis and Simulation (G.P. Galdi et al., eds), pp. 275-332, Birkhäuser, Basel, 2007.
[76]	R. Rannacher, Adaptive Finite Element Discretization of Flow Problems for Goal-Oriented Model Reduction, in Computational Fluid Dynamics Review 2010 (M. Hafez et al., eds), pp. 51-70, World Scientic, Singapore, 2010.doi: 10.1142/9789814313377_0003.
[77]	R. Rannacher, Adaptive Finite Element Methods for Partial Differential Equations, in NSF/CBMS Regional Conference Series in the Mathematical Sciences, College Station, TX 77843, May 18-22, 2009, to appear.
[78]	R. Rannacher, B. Vexler and W. Wollner, A posteriori error estimation in PDE-constrained optimization with pointwise inequality constraints, in Themenband SPP1253 "PDE-Constrained Optimization,'' Springer, Berlin-Heidelberg-New York, 2012, to appear.
[79]	R. Rannacher and S. Turek, Simple nonconforming quadrilateral Stokes element, Numer. Meth. Part. Diff. Equ., 8 (1992), 97-111.
[80]	R. Rannacher and J. Vihharev, Balancing discretization and iteration error in finite element a posteriori error analysis, in Proc. "Computational Analysis and Optimization,'' CAO2011 ECCOMAS Thematic Conference, Jyväskylä, Finland, Springer, Berlin-Heidelberg-New York, 2012, to appear.
[81]	R. Rannacher, A. Westenberger and W. Wollner, Adaptive finite element solution of eigenvalue problems: balancing of discretization and iteration error, J. Numer. Math., 18 (2010), 303-327.doi: 10.1515/jnum.2010.015.
[82]	R. Rautmann, On optimal regularity of Navier-Stokes solutions at time $\,t=0\,$, Math. Z., 184 (1983), 141-149.doi: 10.1007/BF01252853.
[83]	Y. Saad, "Iterative Methods for Sparse Linear Systems," PWS Publishing Company, 1996.
[84]	M. Schäfer and S. Turek, The benchmark problem flow around a cylinder, in Flow Simulation with High-Performance Computers (E. H. Hirschel, ed.), NNFM, Vol. 52, Vieweg, Braunschweig, 1996, 547-566.
[85]	M. Schmich and B. Vexler, Adaptivity with dynamic meshes for space-time finite element discretizations of parabolic equations, SIAM J. Sci. Comput., 30 (2008), 369-393.doi: 10.1137/060670468.
[86]	J. Shen, On error estimates of projection methods for the Navier-Stokes Equations: First order schemes, SIAM J. Numer. Anal., 29 (1992), 57-77.
[87]	J. Shen, On error estimates of the projection methods for the Navier-Stokes Equations: 2nd order schemes, Math. Comp., 65 (1996), 1039-1065.doi: 10.1090/S0025-5718-96-00750-8.
[88]	R. Temam, "Navier-Stokes Equations. Theory and Numerical Analysis," North Holland, Amsterdam, 1977.
[89]	V. Thomée, "Galerkin Finite Element Methods for Parabolic Problems," Springer, Berlin-Heidelberg-New York, 1997.
[90]	L. Tobiska and R. Verfürth, Analysis of a streamline diffusion finite element method for the Stokes and Navier-Stokes equations, SIAM J. Numer. Anal., 33 (1996), 107-127.doi: 10.1137/0733007.
[91]	L. N. Trefethen, Computation of pseudospectra, Acta Numerica, 8 (1999), 247-295.doi: 10.1017/S0962492900002932.
[92]	L. N. Trefethen and M. Embree:, "Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators," Princeton University Press, Princeton and Oxford, 2005.
[93]	L. N. Trefethen, A. E. Trefethen, S. C. Reddy and T. A. Driscoll, A new direction in hydrodynamical stability: Beyond eigenvalues, Tech. Report CTC92TR115 12/92, Cornell Theory Center, Cornell University, 1992.
[94]	L. N. Trefethen, S. C. Reddy and T. A. Driscoll, Hydrodynamic Stability without eigenvalues, Science, New Series, 261 (1993), 578-584.
[95]	S. Turek, A comparative study of some time-stepping techniques for the incompressible Navier-Stokes equations, Int. J. Numer. Meth. Fluids, 22 (1996), 987-1011.doi: 10.1002/(SICI)1097-0363(19960530)22:10<987::AID-FLD394>3.0.CO;2-7.
[96]	S. Turek, On discrete projection methods for the incompressible Navier-Stokes equations: An algorithmic approach, Comput. Methods Appl. Mech. Engrg., 143 (1997), 271-288.doi: 10.1016/S0045-7825(96)01155-3.
[97]	S. Turek, "Efficient Solvers for Incompressible Flow Problems," NNCE, Vol.6, Springer, Berlin-Heidelberg-New York, 1999.
[98]	S. P. Vanka, Block-implicit multigrid solution of Navier-Stokes equations in primitive variables, J. Comp. Phys., 65 (1986), 138-158.doi: 10.1016/0021-9991(86)90008-2.
[99]	J. Van Kan, A second-order accurate pressure-correction scheme for viscous incompressible flow, J. Sci. Stat. Comp., 7 (1986), 870-891.doi: 10.1137/0907059.
[100]	R. Verfürth, "A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques," Wiley/Teubner, New York-Stuttgart, 1996.
[101]	B. Vexler, "Adaptive Finite Element Methods for Parameter Identification Problems," Dissertation, Institute of Applied Mathematics, University of Heidelberg, 2004.
[102]	W. Wollner, "Adaptive Methods for PDE-based Optimal Control with Pointwise Inequality Constraints," doctoral thesis, University of Heidelberg, 2010.