doi: 10.3934/era.2020113

Continuous data assimilation applied to a velocity-vorticity formulation of the 2D Navier-Stokes equations

1. 

School of Mathematical and Statistical Sciences, Clemson University, Clemson, SC, 29634, USA

2. 

Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE, 68588, USA

Received  June 2020 Revised  August 2020 Published  October 2020

Fund Project: The second author is supported by NSF Grants DMS 1716801 and CMMI 1953346. The third and fourth authors are supported by NSF Grant DMS 2011490

We study a continuous data assimilation (CDA) algorithm for a velocity-vorticity formulation of the 2D Navier-Stokes equations in two cases: nudging applied to the velocity and vorticity, and nudging applied to the velocity only. We prove that under a typical finite element spatial discretization and backward Euler temporal discretization, application of CDA preserves the unconditional long-time stability property of the velocity-vorticity method and provides optimal long-time accuracy. These properties hold if nudging is applied only to the velocity, and if nudging is also applied to the vorticity then the optimal long-time accuracy is achieved more rapidly in time. Numerical tests illustrate the theory, and show its effectiveness on an application problem of channel flow past a flat plate.

Citation: Matthew Gardner, Adam Larios, Leo G. Rebholz, Duygu Vargun, Camille Zerfas. Continuous data assimilation applied to a velocity-vorticity formulation of the 2D Navier-Stokes equations. Electronic Research Archive, doi: 10.3934/era.2020113
References:
[1]

M. AkbasS. Kaya and L. G. Rebholz, On the stability at all times of linearly extrapolated BDF2 timestepping for multiphysics incompressible flow problems, Numer. Methods Partial Differential Equations, 33 (2017), 995-1017.  doi: 10.1002/num.22061.  Google Scholar

[2]

M. Akbas, L. G. Rebholz and C. Zerfas, Optimal vorticity accuracy in an efficient velocity-vorticity method for the 2D Navier-Stokes equations, Calcolo, 55 (2018), Paper No. 3, 29 pp. 1–29. doi: 10.1007/s10092-018-0246-7.  Google Scholar

[3]

D. A. F. AlbanezH. Nussenzveig Lopes and E. S. Titi, Continuous data assimilation for the three-dimensional Navier–Stokes-$\alpha$ model, Asymptotic Anal., 97 (2016), 139-164.  doi: 10.3233/ASY-151351.  Google Scholar

[4]

R. A. Anthes, Data assimilation and initialization of hurricane prediction models, J. Atmos. Sci., 31 (1974), 702-719.  doi: 10.1175/1520-0469(1974)031<0702:DAAIOH>2.0.CO;2.  Google Scholar

[5]

A. AzouaniE. Olson and E. S. Titi, Continuous data assimilation using general interpolant observables, Journal of Nonlinear Science, 24 (2014), 277-304.  doi: 10.1007/s00332-013-9189-y.  Google Scholar

[6]

H. BessaihE. Olson and E. S. Titi, Continuous data assimilation with stochastically noisy data, Nonlinearity, 28 (2015), 729-753.  doi: 10.1088/0951-7715/28/3/729.  Google Scholar

[7]

A. BiswasC. FoiasC. F. Mondaini and E. S. Titi, Downscaling data assimilation algorithm with applications to statistical solutions of the Navier–Stokes equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 36 (2019), 295-326.  doi: 10.1016/j.anihpc.2018.05.004.  Google Scholar

[8]

A. BiswasJ. HudsonA. Larios and Y. Pei, Continuous data assimilation for the 2D magnetohydrodynamic equations using one component of the velocity and magnetic fields, Asymptot. Anal., 108 (2018), 1-43.   Google Scholar

[9]

S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Third edition, Texts in Applied Mathematics, 15. Springer, New York, 2008. doi: 10.1007/978-0-387-75934-0.  Google Scholar

[10]

E. Carlson, J. Hudson and A. Larios, Parameter recovery for the 2 dimensional Navier-Stokes equations via continuous data assimilation, SIAM J. Sci. Comput., 42 (2020), A250–A270. doi: 10.1137/19M1248583.  Google Scholar

[11]

E. CelikE. Olson and E. S. Titi, Spectral filtering of interpolant observables for a discrete-in-time downscaling data assimilation algorithm, SIAM J. Appl. Dyn. Syst., 18 (2019), 1118-1142.  doi: 10.1137/18M1218480.  Google Scholar

[12]

T. CharnyiT. HeisterM. A. Olshanskii and L. G. Rebholz, On conservation laws of Navier-Stokes Galerkin discretizations, Journal of Computational Physics, 337 (2017), 289-308.  doi: 10.1016/j.jcp.2017.02.039.  Google Scholar

[13]

P. Clark Di LeoniA. Mazzino and L. Biferale, Synchronization to big-data: Nudging the Navier-Stokes equations for data assimilation of turbulent flows, Physical Review X, 10 (2020), 1-15.   Google Scholar

[14]

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S. DesamsettiI. HoteitO. KnioE. TitiS. Langodan and H. P. Dasari, Efficient dynamical downscaling of general circulation models using continuous data assimilation, Quarterly Journal of the Royal Meteorological Society, 145 (2019), 3175-3194.  doi: 10.1002/qj.3612.  Google Scholar

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A. FarhatN. E. Glatt-HoltzV. R. MartinezS. A. McQuarrie and J. P. Whitehead, Data assimilation in large Prandtl Rayleigh–Bénard convection from thermal measurements, SIAM J. Appl. Dyn. Syst., 19 (2020), 510-540.  doi: 10.1137/19M1248327.  Google Scholar

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A. FarhatM. S. Jolly and E. S. Titi, Continuous data assimilation for the 2D Bénard convection through velocity measurements alone, Phys. D, 303 (2015), 59-66.  doi: 10.1016/j.physd.2015.03.011.  Google Scholar

[19]

A. Farhat, E. Lunasin and E. S. Titi, A data assimilation algorithm: The paradigm of the 3D Leray-α model of turbulence, Partial Differential Equations Arising from Physics and Geometry, London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge, 450 (2019), 253-273.  Google Scholar

[20]

C. FoiasC. F. Mondaini and E. S. Titi, A discrete data assimilation scheme for the solutions of the two-dimensional Navier-Stokes equations and their statistics, SIAM J. Appl. Dyn. Syst., 15 (2016), 2109-2142.  doi: 10.1137/16M1076526.  Google Scholar

[21]

B. Garcia-ArchillaJ. Novo and E. S. Titi, Uniform in time error estimates for a finite element method applied to a downscaling data assimilation algorithm for the Navier-Stokes equations, SIAM Journal on Numerical Analysis, 58 (2020), 410-429.  doi: 10.1137/19M1246845.  Google Scholar

[22]

M. GeshoE. Olson and E. S. Titi, A computational study of a data assimilation algorithm for the two-dimensional Navier–Stokes equations, Commun. Comput. Phys., 19 (2016), 1094-1110.  doi: 10.4208/cicp.060515.161115a.  Google Scholar

[23]

P. Gresho and R. Sani, Incompressible Flow and the Finite Element Method, Vol. 2, Wiley, 1998. Google Scholar

[24]

J. Guzman and L. R. Scott, The Scott-Vogelius finite elements revisited, Math. Comp., 88 (2019), 515-529.  doi: 10.1090/mcom/3346.  Google Scholar

[25]

T. HeisterM. A. Olshanskii and L. G. Rebholz, Unconditional long-time stability of a velocity-vorticity method for the 2D Navier-Stokes equations, Numer. Math., 135 (2017), 143-167.  doi: 10.1007/s00211-016-0794-1.  Google Scholar

[26]

J. E. Hoke and R. A. Anthes, The initialization of numerical models by a dynamic-initialization technique, Monthly Weather Review, 104 (1976), 1551-1556.  doi: 10.1175/1520-0493(1976)104<1551:TIONMB>2.0.CO;2.  Google Scholar

[27]

H. A. Ibdah, C. F. Mondaini and E. S. Titi, Fully discrete numerical schemes of a data assimilation algorithm: Uniform-in-time error estimates, IMA Journal of Numerical Analysis, Drz043, (2019). doi: 10.1093/imanum/drz043.  Google Scholar

[28]

N. Jiang, A second order ensemble method based on a blended BDF time-stepping scheme for time dependent Navier-Stokes equations, Numerical Methods for Partial Differential Equations, 33 (2017), 34-61.  doi: 10.1002/num.22070.  Google Scholar

[29]

R. E. Kalman, A new approach to linear filtering and prediction problems, Trans. ASME Ser. D. J. Basic Engrg., 82 (1960), 35-45.  doi: 10.1115/1.3662552.  Google Scholar

[30]

A. LariosL. G. Rebholz and C. Zerfas, Global in time stability and accuracy of IMEX-FEM data assimilation schemes for Navier-Stokes equations, Computer Methods in Applied Mechanics and Engineering, 345 (2019), 1077-1093.  doi: 10.1016/j.cma.2018.09.004.  Google Scholar

[31]

A. Larios and C. Victor, Continuous data assimilation with a moving cluster of data points for a reaction diffusion equation: A computational study, Commun. Comp. Phys., (accepted for publication). Google Scholar

[32]

K. Law, A. Stuart and K. Zygalakis, A Mathematical Introduction to Data Assimilation, Texts in Applied Mathematics, 62. Springer, Cham, 2015. doi: 10.1007/978-3-319-20325-6.  Google Scholar

[33]

W. LaytonC. C. ManicaM. NedaM. Olshanskii and L. G. Rebholz, On the accuracy of the rotation form in simulations of the Navier-Stokes equations, Journal of Computational Physics, 228 (2009), 3433-3447.  doi: 10.1016/j.jcp.2009.01.027.  Google Scholar

[34]

H. K. LeeM. A. Olshanskii and L. G. Rebholz, On error analysis for the 3D Navier-Stokes equations in velocity-vorticity-helicity form, SIAM Journal on Numerical Analysis, 49 (2011), 711-732.  doi: 10.1137/10080124X.  Google Scholar

[35]

C. F. Mondaini and E. S. Titi, Uniform-in-time error estimates for the postprocessing Galerkin method applied to a data assimilation algorithm, SIAM J. Numer. Anal., 56 (2018), 78-110.  doi: 10.1137/16M110962X.  Google Scholar

[36]

M. A. OlshanskiiT. HeisterL. G. Rebholz and K. J. Galvin, Natural vorticity boundary conditions on solid walls, Computer Methods in Applied Mechanics and Engineering, 297 (2015), 18-37.  doi: 10.1016/j.cma.2015.08.011.  Google Scholar

[37]

M. A. Olshanskii and L. G. Rebholz, Velocity-vorticity-helicity formulation and a solver for the Navier-Stokes equations, Journal of Computational Physics, 229 (2010), 4291-4303.  doi: 10.1016/j.jcp.2010.02.012.  Google Scholar

[38]

M. A. OlshanskiiL. G. Rebholz and A. J. Salgado, On well-posedness of a velocity-vorticity formulation of the Navier-Stokes equations with no-slip boundary conditions, Discrete Contin. Dyn. Syst., 38 (2018), 3459-3477.  doi: 10.3934/dcds.2018148.  Google Scholar

[39]

M. A. Olshanskii and A. Reusken, Grad-div stabilization for the Stokes equations, Math. Comp., 73 (2004), 1699-1718.  doi: 10.1090/S0025-5718-03-01629-6.  Google Scholar

[40]

Y. Pei, Continuous data assimilation for the 3D primitive equations of the ocean, Comm. Pure Appl. Math., 18 (2019), 643-661.  doi: 10.3934/cpaa.2019032.  Google Scholar

[41]

L. Rebholz and C. Zerfas, Simple and efficient continuous data assimilation of evolution equations via algebraic nudging, Submitted. Google Scholar

[42]

P. W. SchroederC. LehrenfeldA. Linke and G. Lube, Towards computable flows and robust estimates for inf-sup stable fem applied to the time dependent incompressible Navier-Stokes equations, SeMA J., 75 (2018), 629-653.  doi: 10.1007/s40324-018-0157-1.  Google Scholar

[43]

C. Zerfas, Numerical Methods and Analysis for Continuous Data Assimilation in Fluid Models, PhD thesis, Clemson University, 2019,132 pp, https://tigerprints.clemson.edu/all_dissertations/2428.  Google Scholar

[44]

C. Zerfas, L. G. Rebholz, M. Schneier and T. Iliescu, Continuous data assimilation reduced order models of fluid flow, Computer Methods in Applied Mechanics and Engineering, 357 (2019), 112596, 18 pp. doi: 10.1016/j.cma.2019.112596.  Google Scholar

show all references

References:
[1]

M. AkbasS. Kaya and L. G. Rebholz, On the stability at all times of linearly extrapolated BDF2 timestepping for multiphysics incompressible flow problems, Numer. Methods Partial Differential Equations, 33 (2017), 995-1017.  doi: 10.1002/num.22061.  Google Scholar

[2]

M. Akbas, L. G. Rebholz and C. Zerfas, Optimal vorticity accuracy in an efficient velocity-vorticity method for the 2D Navier-Stokes equations, Calcolo, 55 (2018), Paper No. 3, 29 pp. 1–29. doi: 10.1007/s10092-018-0246-7.  Google Scholar

[3]

D. A. F. AlbanezH. Nussenzveig Lopes and E. S. Titi, Continuous data assimilation for the three-dimensional Navier–Stokes-$\alpha$ model, Asymptotic Anal., 97 (2016), 139-164.  doi: 10.3233/ASY-151351.  Google Scholar

[4]

R. A. Anthes, Data assimilation and initialization of hurricane prediction models, J. Atmos. Sci., 31 (1974), 702-719.  doi: 10.1175/1520-0469(1974)031<0702:DAAIOH>2.0.CO;2.  Google Scholar

[5]

A. AzouaniE. Olson and E. S. Titi, Continuous data assimilation using general interpolant observables, Journal of Nonlinear Science, 24 (2014), 277-304.  doi: 10.1007/s00332-013-9189-y.  Google Scholar

[6]

H. BessaihE. Olson and E. S. Titi, Continuous data assimilation with stochastically noisy data, Nonlinearity, 28 (2015), 729-753.  doi: 10.1088/0951-7715/28/3/729.  Google Scholar

[7]

A. BiswasC. FoiasC. F. Mondaini and E. S. Titi, Downscaling data assimilation algorithm with applications to statistical solutions of the Navier–Stokes equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 36 (2019), 295-326.  doi: 10.1016/j.anihpc.2018.05.004.  Google Scholar

[8]

A. BiswasJ. HudsonA. Larios and Y. Pei, Continuous data assimilation for the 2D magnetohydrodynamic equations using one component of the velocity and magnetic fields, Asymptot. Anal., 108 (2018), 1-43.   Google Scholar

[9]

S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Third edition, Texts in Applied Mathematics, 15. Springer, New York, 2008. doi: 10.1007/978-0-387-75934-0.  Google Scholar

[10]

E. Carlson, J. Hudson and A. Larios, Parameter recovery for the 2 dimensional Navier-Stokes equations via continuous data assimilation, SIAM J. Sci. Comput., 42 (2020), A250–A270. doi: 10.1137/19M1248583.  Google Scholar

[11]

E. CelikE. Olson and E. S. Titi, Spectral filtering of interpolant observables for a discrete-in-time downscaling data assimilation algorithm, SIAM J. Appl. Dyn. Syst., 18 (2019), 1118-1142.  doi: 10.1137/18M1218480.  Google Scholar

[12]

T. CharnyiT. HeisterM. A. Olshanskii and L. G. Rebholz, On conservation laws of Navier-Stokes Galerkin discretizations, Journal of Computational Physics, 337 (2017), 289-308.  doi: 10.1016/j.jcp.2017.02.039.  Google Scholar

[13]

P. Clark Di LeoniA. Mazzino and L. Biferale, Synchronization to big-data: Nudging the Navier-Stokes equations for data assimilation of turbulent flows, Physical Review X, 10 (2020), 1-15.   Google Scholar

[14]

R. Daley, Atmospheric Data Analysis, Cambridge Atmospheric and Space Science Series, Cambridge University Press, 1993. doi: 10.4267/2042/51948.  Google Scholar

[15]

S. DesamsettiI. HoteitO. KnioE. TitiS. Langodan and H. P. Dasari, Efficient dynamical downscaling of general circulation models using continuous data assimilation, Quarterly Journal of the Royal Meteorological Society, 145 (2019), 3175-3194.  doi: 10.1002/qj.3612.  Google Scholar

[16]

A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements, Applied Mathematical Sciences, 159. Springer-Verlag, New York, 2004. doi: 10.1007/978-1-4757-4355-5.  Google Scholar

[17]

A. FarhatN. E. Glatt-HoltzV. R. MartinezS. A. McQuarrie and J. P. Whitehead, Data assimilation in large Prandtl Rayleigh–Bénard convection from thermal measurements, SIAM J. Appl. Dyn. Syst., 19 (2020), 510-540.  doi: 10.1137/19M1248327.  Google Scholar

[18]

A. FarhatM. S. Jolly and E. S. Titi, Continuous data assimilation for the 2D Bénard convection through velocity measurements alone, Phys. D, 303 (2015), 59-66.  doi: 10.1016/j.physd.2015.03.011.  Google Scholar

[19]

A. Farhat, E. Lunasin and E. S. Titi, A data assimilation algorithm: The paradigm of the 3D Leray-α model of turbulence, Partial Differential Equations Arising from Physics and Geometry, London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge, 450 (2019), 253-273.  Google Scholar

[20]

C. FoiasC. F. Mondaini and E. S. Titi, A discrete data assimilation scheme for the solutions of the two-dimensional Navier-Stokes equations and their statistics, SIAM J. Appl. Dyn. Syst., 15 (2016), 2109-2142.  doi: 10.1137/16M1076526.  Google Scholar

[21]

B. Garcia-ArchillaJ. Novo and E. S. Titi, Uniform in time error estimates for a finite element method applied to a downscaling data assimilation algorithm for the Navier-Stokes equations, SIAM Journal on Numerical Analysis, 58 (2020), 410-429.  doi: 10.1137/19M1246845.  Google Scholar

[22]

M. GeshoE. Olson and E. S. Titi, A computational study of a data assimilation algorithm for the two-dimensional Navier–Stokes equations, Commun. Comput. Phys., 19 (2016), 1094-1110.  doi: 10.4208/cicp.060515.161115a.  Google Scholar

[23]

P. Gresho and R. Sani, Incompressible Flow and the Finite Element Method, Vol. 2, Wiley, 1998. Google Scholar

[24]

J. Guzman and L. R. Scott, The Scott-Vogelius finite elements revisited, Math. Comp., 88 (2019), 515-529.  doi: 10.1090/mcom/3346.  Google Scholar

[25]

T. HeisterM. A. Olshanskii and L. G. Rebholz, Unconditional long-time stability of a velocity-vorticity method for the 2D Navier-Stokes equations, Numer. Math., 135 (2017), 143-167.  doi: 10.1007/s00211-016-0794-1.  Google Scholar

[26]

J. E. Hoke and R. A. Anthes, The initialization of numerical models by a dynamic-initialization technique, Monthly Weather Review, 104 (1976), 1551-1556.  doi: 10.1175/1520-0493(1976)104<1551:TIONMB>2.0.CO;2.  Google Scholar

[27]

H. A. Ibdah, C. F. Mondaini and E. S. Titi, Fully discrete numerical schemes of a data assimilation algorithm: Uniform-in-time error estimates, IMA Journal of Numerical Analysis, Drz043, (2019). doi: 10.1093/imanum/drz043.  Google Scholar

[28]

N. Jiang, A second order ensemble method based on a blended BDF time-stepping scheme for time dependent Navier-Stokes equations, Numerical Methods for Partial Differential Equations, 33 (2017), 34-61.  doi: 10.1002/num.22070.  Google Scholar

[29]

R. E. Kalman, A new approach to linear filtering and prediction problems, Trans. ASME Ser. D. J. Basic Engrg., 82 (1960), 35-45.  doi: 10.1115/1.3662552.  Google Scholar

[30]

A. LariosL. G. Rebholz and C. Zerfas, Global in time stability and accuracy of IMEX-FEM data assimilation schemes for Navier-Stokes equations, Computer Methods in Applied Mechanics and Engineering, 345 (2019), 1077-1093.  doi: 10.1016/j.cma.2018.09.004.  Google Scholar

[31]

A. Larios and C. Victor, Continuous data assimilation with a moving cluster of data points for a reaction diffusion equation: A computational study, Commun. Comp. Phys., (accepted for publication). Google Scholar

[32]

K. Law, A. Stuart and K. Zygalakis, A Mathematical Introduction to Data Assimilation, Texts in Applied Mathematics, 62. Springer, Cham, 2015. doi: 10.1007/978-3-319-20325-6.  Google Scholar

[33]

W. LaytonC. C. ManicaM. NedaM. Olshanskii and L. G. Rebholz, On the accuracy of the rotation form in simulations of the Navier-Stokes equations, Journal of Computational Physics, 228 (2009), 3433-3447.  doi: 10.1016/j.jcp.2009.01.027.  Google Scholar

[34]

H. K. LeeM. A. Olshanskii and L. G. Rebholz, On error analysis for the 3D Navier-Stokes equations in velocity-vorticity-helicity form, SIAM Journal on Numerical Analysis, 49 (2011), 711-732.  doi: 10.1137/10080124X.  Google Scholar

[35]

C. F. Mondaini and E. S. Titi, Uniform-in-time error estimates for the postprocessing Galerkin method applied to a data assimilation algorithm, SIAM J. Numer. Anal., 56 (2018), 78-110.  doi: 10.1137/16M110962X.  Google Scholar

[36]

M. A. OlshanskiiT. HeisterL. G. Rebholz and K. J. Galvin, Natural vorticity boundary conditions on solid walls, Computer Methods in Applied Mechanics and Engineering, 297 (2015), 18-37.  doi: 10.1016/j.cma.2015.08.011.  Google Scholar

[37]

M. A. Olshanskii and L. G. Rebholz, Velocity-vorticity-helicity formulation and a solver for the Navier-Stokes equations, Journal of Computational Physics, 229 (2010), 4291-4303.  doi: 10.1016/j.jcp.2010.02.012.  Google Scholar

[38]

M. A. OlshanskiiL. G. Rebholz and A. J. Salgado, On well-posedness of a velocity-vorticity formulation of the Navier-Stokes equations with no-slip boundary conditions, Discrete Contin. Dyn. Syst., 38 (2018), 3459-3477.  doi: 10.3934/dcds.2018148.  Google Scholar

[39]

M. A. Olshanskii and A. Reusken, Grad-div stabilization for the Stokes equations, Math. Comp., 73 (2004), 1699-1718.  doi: 10.1090/S0025-5718-03-01629-6.  Google Scholar

[40]

Y. Pei, Continuous data assimilation for the 3D primitive equations of the ocean, Comm. Pure Appl. Math., 18 (2019), 643-661.  doi: 10.3934/cpaa.2019032.  Google Scholar

[41]

L. Rebholz and C. Zerfas, Simple and efficient continuous data assimilation of evolution equations via algebraic nudging, Submitted. Google Scholar

[42]

P. W. SchroederC. LehrenfeldA. Linke and G. Lube, Towards computable flows and robust estimates for inf-sup stable fem applied to the time dependent incompressible Navier-Stokes equations, SeMA J., 75 (2018), 629-653.  doi: 10.1007/s40324-018-0157-1.  Google Scholar

[43]

C. Zerfas, Numerical Methods and Analysis for Continuous Data Assimilation in Fluid Models, PhD thesis, Clemson University, 2019,132 pp, https://tigerprints.clemson.edu/all_dissertations/2428.  Google Scholar

[44]

C. Zerfas, L. G. Rebholz, M. Schneier and T. Iliescu, Continuous data assimilation reduced order models of fluid flow, Computer Methods in Applied Mechanics and Engineering, 357 (2019), 112596, 18 pp. doi: 10.1016/j.cma.2019.112596.  Google Scholar

Figure 1.  Shown above are $ L^2 $ velocity and vorticity errors for Algorithm 3.6 with $ \mu_1 = \mu_2 = \mu $, with varying $ \mu>0 $
Figure 2.  Shown above are $ L^2 $ velocity and vorticity errors (from left to right) for Algorithm 3.6 with varying $ \mu_1 $ and $ \mu_2 = 0 $
Figure 3.  Setup for the flow past a normal flat plate
Figure 4.  $ L^2 $ velocity and vorticity errors (from left to right) for Algorithm 3.6 with $ \mu_1 = \mu_2 = \mu >0 $ (top) and $ \mu_1 = \mu>0, \mu_2 = 0 $ (bottom)
Figure 5.  Contour plots of velocity for DNS (left), VV-DA with $ \mu_1 = \mu_2 = 10 $ (center), and their difference (right), for times $ t = 0,\ 0.1,\ 1,\ 10,\ 20,\ 80 $ (top to bottom)
Figure 6.  Contour plots of vorticity for DNS (left), VV-DA with $ \mu_1 = \mu_2 = 10 $ (center), and their difference (right), for times $ t = 0,\ 0.1,\ 1,\ 10,\ 20,\ 80 $ (top to bottom)
Figure 7.  Contour plots of velocity for DNS (left), VV-DA with $ \mu_1 = 10,\ \mu_2 = 0 $ (center), and their difference (right), for times $ t = 0,\ 0.1,\ 1,\ 10,\ 20,\ 80 $ (top to bottom)
Figure 8.  Contour plots of vorticity for DNS (left), VV-DA with $ \mu_1 = 10,\ \mu_2 = 0 $ (center), and their difference (right), for times $ t = 0,\ 0.1,\ 1,\ 10,\ 20,\ 80 $ (top to bottom)
Figure 9.  $ L^2 $ velocity and vorticity errors (from left to right) for Algorithm 3.6 with $ \mu_1 = \mu_2 = \mu >0 $ (top) and $ \mu_1 = \mu>0, \mu_2 = 0 $ (bottom), with $ Re = 100 $
Table 1.  Shown above are $ L^2 $ velocity and vorticity errors and convergence rates on varying mesh widths, at the final time $ T = 1 $, using Algorithm 3.6 with $ \mu_1 = \mu_2 = 100 $
h $ \|e_v(T) \| $ rate $ \|e_w(T)\| $ rate
1/4 2.62008e-03 - 7.70647e-03 -
1/8 3.20467e-04 3.0314 9.68456e-04 2.9923
1/16 3.97307e-05 3.0146 1.20888e-04 3.0041
1/32 4.94529e-06 3.0061 1.50809e-05 3.0029
1/64 6.19332e-07 2.9973 1.99325e-06 2.9195
1/128 8.13141e-08 2.9247 3.15236e-07 2.5855
h $ \|e_v(T) \| $ rate $ \|e_w(T)\| $ rate
1/4 2.62008e-03 - 7.70647e-03 -
1/8 3.20467e-04 3.0314 9.68456e-04 2.9923
1/16 3.97307e-05 3.0146 1.20888e-04 3.0041
1/32 4.94529e-06 3.0061 1.50809e-05 3.0029
1/64 6.19332e-07 2.9973 1.99325e-06 2.9195
1/128 8.13141e-08 2.9247 3.15236e-07 2.5855
Table 2.  Shown above are $ L^2 $ velocity and vorticity errors and convergence rates on varying mesh widths, at the final time $ T = 1 $, using Algorithm 3.6 with $ \mu_1 = 100 $ and $ \mu_2 = 0 $
h $ \|e_v(T)\| $ rate $ \|e_w(T)\| $ rate
1/4 2.62003e-03 - 7.79431e-03 -
1/8 3.20466e-04 3.0313 9.70492e-04 3.0056
1/16 3.97175e-05 3.0123 1.20897e-04 3.0049
1/32 4.94501e-06 3.0057 1.50883e-05 3.0023
1/64 6.17406e-07 3.0017 2.08215e-06 2.8573
1/128 8.11244e-08 2.9280 9.37122e-07 1.1518
h $ \|e_v(T)\| $ rate $ \|e_w(T)\| $ rate
1/4 2.62003e-03 - 7.79431e-03 -
1/8 3.20466e-04 3.0313 9.70492e-04 3.0056
1/16 3.97175e-05 3.0123 1.20897e-04 3.0049
1/32 4.94501e-06 3.0057 1.50883e-05 3.0023
1/64 6.17406e-07 3.0017 2.08215e-06 2.8573
1/128 8.11244e-08 2.9280 9.37122e-07 1.1518
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