# American Institute of Mathematical Sciences

August  2021, 29(3): 2223-2247. 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, 2021, 29 (3) : 2223-2247. doi: 10.3934/era.2020113
##### References:
 [1] M. Akbas, S. 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. Albanez, H. 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. Azouani, E. 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. Bessaih, E. 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. Biswas, C. Foias, C. 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. Biswas, J. Hudson, A. 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. Celik, E. 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. Charnyi, T. Heister, M. 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 Leoni, A. 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. Desamsetti, I. Hoteit, O. Knio, E. Titi, S. 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. Farhat, N. E. Glatt-Holtz, V. R. Martinez, S. 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. Farhat, M. 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. Foias, C. 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-Archilla, J. 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. Gesho, E. 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. Heister, M. 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. Larios, L. 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. Layton, C. C. Manica, M. Neda, M. 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. Lee, M. 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. Olshanskii, T. Heister, L. 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. Olshanskii, L. 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. Schroeder, C. Lehrenfeld, A. 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. Akbas, S. 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. Albanez, H. 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. Azouani, E. 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. Bessaih, E. 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. Biswas, C. Foias, C. 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. Biswas, J. Hudson, A. 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. Celik, E. 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. Charnyi, T. Heister, M. 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 Leoni, A. 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. Desamsetti, I. Hoteit, O. Knio, E. Titi, S. 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. Farhat, N. E. Glatt-Holtz, V. R. Martinez, S. 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. Farhat, M. 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. Foias, C. 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-Archilla, J. 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. Gesho, E. 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. Heister, M. 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. Larios, L. 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. Layton, C. C. Manica, M. Neda, M. 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. Lee, M. 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. Olshanskii, T. Heister, L. 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. Olshanskii, L. 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. Schroeder, C. Lehrenfeld, A. 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
Shown above are $L^2$ velocity and vorticity errors for Algorithm 3.6 with $\mu_1 = \mu_2 = \mu$, with varying $\mu>0$
Shown above are $L^2$ velocity and vorticity errors (from left to right) for Algorithm 3.6 with varying $\mu_1$ and $\mu_2 = 0$
Setup for the flow past a normal flat plate
$L^2$ 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)
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)
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)
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)
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)
$L^2$ 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 $Re = 100$
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
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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