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On well-posedness of a velocity-vorticity formulation of the stationary Navier-Stokes equations with no-slip boundary conditions
1. | Department of Mathematics, University of Houston, Houston, TX, 77204, USA |
2. | Department of Mathematical Sciences, Clemson University, Clemson, SC, 29634, USA |
3. | Department of Mathematics, University of Tennessee, Knoxville, TN 37996, USA |
We study well-posedness of a velocity-vorticity formulation of the Navier-Stokes equations, supplemented with no-slip velocity boundary conditions, a corresponding zero-normal condition for vorticity on the boundary, along with a natural vorticity boundary condition depending on a pressure functional. In the stationary case we prove existence and uniqueness of a suitable weak solution to the system under a small data condition. The topic of the paper is driven by recent developments of vorticity based numerical methods for the Navier-Stokes equations.
References:
[1] |
R. A. Adams,
Sobolev Spaces, Academic Press, New York, 1975. |
[2] |
M. Akbas, L. Rebholz and C. Zerfas, Optimal vorticity accuracy in an efficient velocity-vorticity method for the 2D Navier-Stokes equations,
Calcolo, 55 (2018), p3.
doi: 10.1007/s10092-018-0246-7. |
[3] |
C. Begue, C. Conca, F. Murat and O. Pironneau, Les equations de Stokes et de Navier-Stokes
avec des conditions aux limites sur la pression, in Nonlinear Partial Differential Equations
and their Applications, Pitman Research Notes in Math. , (eds. H. Brezis and J. L. Lions),
College de France Seminar, 181 (1988), 179–264. |
[4] |
M. Benzi, M. A. Olshanskii, L. G. Rebholz and Z. Wang,
Assessment of a vorticity based solver for the Navier-Stokes equations, Computer Methods in Applied Mechanics and Engineering, 247 (2012), 216-225.
doi: 10.1016/j.cma.2012.07.016. |
[5] |
W. Borchers and H. Sohr,
On the equations rot v = g and div u = f with zero boundary conditions, Hokkaido Math. J, 19 (1990), 67-87.
doi: 10.14492/hokmj/1381517172. |
[6] |
S. Charnyi, T. Heister, M. Olshanskii and L. 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. |
[7] |
P. G. Ciarlet,
Linear and Nonlinear Functional Analysis with Applications, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2013. |
[8] |
A. Ern and J. -L. Guermond,
Theory and Practice of Finite Elements, vol. 159, Applied Mathematical Sciences, 159. Springer-Verlag, New York, 2004.
doi: 10.1007/978-1-4757-4355-5. |
[9] |
L. C. Evans,
Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics, 2nd edition, American Mathematical Society, Providence, RI, 2010, URL http://dx.doi.org/10.1090/gsm/019.
doi: 10.1090/gsm/019. |
[10] |
G. P. Galdi,
An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Steady-state Problems, Springer Science & Business Media, 2011.
doi: 10.1007/978-0-387-09620-9. |
[11] |
T. B. Gatski,
Review of incompressible fluid flow computations using the vorticity-velocity formulation, Appl. Numer. Math., 7 (1991), 227-239.
doi: 10.1016/0168-9274(91)90035-X. |
[12] |
D. Gilbarg and N. S. Trudinger,
Elliptic Partial Differential Equations of Second Order, Springer-Verlag, New York, 1983.
doi: 10.1007/978-3-642-61798-0. |
[13] |
V. Girault, Curl-conforming finite element methods for Navier-Stokes equations with nonstandard boundary conditions in $\textbf{R}^3$, The Navier-Stokes Equations (Oberwolfach, 1988), 201–218,
Lecture Notes in Math., 1431, Springer, Berlin, 1990.
doi: 10.1007/BFb0086071. |
[14] |
V. Girault and P. -A. Raviart,
Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms, Springer-Verlag, 1986.
doi: 10.1007/978-3-642-61623-5. |
[15] |
P. Gresho and R. Sani, Incompressible Flow and the Finite Element {M}ethod, vol. 2, Wiley, 1998. |
[16] |
P. Grisvard,
Elliptic Problems in Nonsmooth Domains, SIAM, Philadelphia, 2011.
doi: 10.1137/1.9781611972030.ch1. |
[17] |
G. Guevremont, W. G. Habashi and M. M. Hafez, Finite element solution of the Navier-Stokes equations by a velocity-vorticity method, Int. J. Numer. Methods Fluids, 10 (1990), 461-475. |
[18] |
M. D. Gunzburger,
Finite Element Methods for Viscous Incompressible Flows: A Guide to Theory, Practice, and Algorithm, Academic Press Inc., Boston, 1989. |
[19] |
T. Heister, M. A. Olshanskii and L. G. Rebholz,
Unconditional long-time stability of velocity-vorticity method for 2D Navier-Stokes equations, Numerische Mathematik, 135 (2017), 143-167.
doi: 10.1007/s00211-016-0794-1. |
[20] |
W. Layton, Introduction to Finite Element Methods for Incompressible, Viscous Flow, SIAM, Philadelphia, 2008. |
[21] |
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 J. Numer. Anal., 49 (2011), 711-732.
doi: 10.1137/10080124X. |
[22] |
D. C. Lo, D. L. Young and K. Murugesan,
An accurate numerical solution algorithm for 3D velocity-vorticity Navier-Stokes equations by the {DQ} method, Commun. Numer. Meth. Engng, 22 (2006), 235-250.
doi: 10.1002/cnm.817. |
[23] |
A. J. Majda and A. L. Bertozzi,
Vorticity and Incompressible Flow, vol. 27, Cambridge University Press, 2002. |
[24] |
H. L. Meitz and H. F. Fasel,
A compact-difference scheme for the Navier-Stokes equations in vorticity-velocity formulation, J. Comput. Phys., 157 (2000), 371-403.
doi: 10.1006/jcph.1999.6387. |
[25] |
M. A. Olshanskii, T. Heister, L. Rebholz and K. 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. |
[26] |
M. A. Olshanskii and L. G. Rebholz,
Velocity-vorticity-helicity formulation and a solver for the {N}avier-{S}tokes equations, Journal of Computational Physics, 229 (2010), 4291-4303.
doi: 10.1016/j.jcp.2010.02.012. |
[27] |
A. Palha and M. Gerritsma,
A mass, energy, enstrophy and vorticity conserving (MEEVC) mimetic spectral element discretization for the 2D incompressible Navier-Stokes equations, Journal of Computational Physics, 328 (2017), 200-220.
doi: 10.1016/j.jcp.2016.10.009. |
[28] |
R. Temam,
Navier-Stokes Equations: Theory and Numerical Analysis, North Holland Publishing Company, New York, 1977. |
[29] |
K. L. Wong and A. J. Baker,
A 3D incompressible Navier-Stokes velocity-vorticity weak form finite element algorithm, Int. J. Numer. Meth. Fluids, 38 (2002), 99-123.
doi: 10.1002/fld.204. |
[30] |
X. H. Wu, J. Z. Wu and J. M. Wu,
Effective vorticity-velocity formulations for the three-dimensional incompressible viscous flows, J. Comput. Phys., 122 (1995), 68-82.
doi: 10.1006/jcph.1995.1197. |
show all references
References:
[1] |
R. A. Adams,
Sobolev Spaces, Academic Press, New York, 1975. |
[2] |
M. Akbas, L. Rebholz and C. Zerfas, Optimal vorticity accuracy in an efficient velocity-vorticity method for the 2D Navier-Stokes equations,
Calcolo, 55 (2018), p3.
doi: 10.1007/s10092-018-0246-7. |
[3] |
C. Begue, C. Conca, F. Murat and O. Pironneau, Les equations de Stokes et de Navier-Stokes
avec des conditions aux limites sur la pression, in Nonlinear Partial Differential Equations
and their Applications, Pitman Research Notes in Math. , (eds. H. Brezis and J. L. Lions),
College de France Seminar, 181 (1988), 179–264. |
[4] |
M. Benzi, M. A. Olshanskii, L. G. Rebholz and Z. Wang,
Assessment of a vorticity based solver for the Navier-Stokes equations, Computer Methods in Applied Mechanics and Engineering, 247 (2012), 216-225.
doi: 10.1016/j.cma.2012.07.016. |
[5] |
W. Borchers and H. Sohr,
On the equations rot v = g and div u = f with zero boundary conditions, Hokkaido Math. J, 19 (1990), 67-87.
doi: 10.14492/hokmj/1381517172. |
[6] |
S. Charnyi, T. Heister, M. Olshanskii and L. 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. |
[7] |
P. G. Ciarlet,
Linear and Nonlinear Functional Analysis with Applications, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2013. |
[8] |
A. Ern and J. -L. Guermond,
Theory and Practice of Finite Elements, vol. 159, Applied Mathematical Sciences, 159. Springer-Verlag, New York, 2004.
doi: 10.1007/978-1-4757-4355-5. |
[9] |
L. C. Evans,
Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics, 2nd edition, American Mathematical Society, Providence, RI, 2010, URL http://dx.doi.org/10.1090/gsm/019.
doi: 10.1090/gsm/019. |
[10] |
G. P. Galdi,
An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Steady-state Problems, Springer Science & Business Media, 2011.
doi: 10.1007/978-0-387-09620-9. |
[11] |
T. B. Gatski,
Review of incompressible fluid flow computations using the vorticity-velocity formulation, Appl. Numer. Math., 7 (1991), 227-239.
doi: 10.1016/0168-9274(91)90035-X. |
[12] |
D. Gilbarg and N. S. Trudinger,
Elliptic Partial Differential Equations of Second Order, Springer-Verlag, New York, 1983.
doi: 10.1007/978-3-642-61798-0. |
[13] |
V. Girault, Curl-conforming finite element methods for Navier-Stokes equations with nonstandard boundary conditions in $\textbf{R}^3$, The Navier-Stokes Equations (Oberwolfach, 1988), 201–218,
Lecture Notes in Math., 1431, Springer, Berlin, 1990.
doi: 10.1007/BFb0086071. |
[14] |
V. Girault and P. -A. Raviart,
Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms, Springer-Verlag, 1986.
doi: 10.1007/978-3-642-61623-5. |
[15] |
P. Gresho and R. Sani, Incompressible Flow and the Finite Element {M}ethod, vol. 2, Wiley, 1998. |
[16] |
P. Grisvard,
Elliptic Problems in Nonsmooth Domains, SIAM, Philadelphia, 2011.
doi: 10.1137/1.9781611972030.ch1. |
[17] |
G. Guevremont, W. G. Habashi and M. M. Hafez, Finite element solution of the Navier-Stokes equations by a velocity-vorticity method, Int. J. Numer. Methods Fluids, 10 (1990), 461-475. |
[18] |
M. D. Gunzburger,
Finite Element Methods for Viscous Incompressible Flows: A Guide to Theory, Practice, and Algorithm, Academic Press Inc., Boston, 1989. |
[19] |
T. Heister, M. A. Olshanskii and L. G. Rebholz,
Unconditional long-time stability of velocity-vorticity method for 2D Navier-Stokes equations, Numerische Mathematik, 135 (2017), 143-167.
doi: 10.1007/s00211-016-0794-1. |
[20] |
W. Layton, Introduction to Finite Element Methods for Incompressible, Viscous Flow, SIAM, Philadelphia, 2008. |
[21] |
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 J. Numer. Anal., 49 (2011), 711-732.
doi: 10.1137/10080124X. |
[22] |
D. C. Lo, D. L. Young and K. Murugesan,
An accurate numerical solution algorithm for 3D velocity-vorticity Navier-Stokes equations by the {DQ} method, Commun. Numer. Meth. Engng, 22 (2006), 235-250.
doi: 10.1002/cnm.817. |
[23] |
A. J. Majda and A. L. Bertozzi,
Vorticity and Incompressible Flow, vol. 27, Cambridge University Press, 2002. |
[24] |
H. L. Meitz and H. F. Fasel,
A compact-difference scheme for the Navier-Stokes equations in vorticity-velocity formulation, J. Comput. Phys., 157 (2000), 371-403.
doi: 10.1006/jcph.1999.6387. |
[25] |
M. A. Olshanskii, T. Heister, L. Rebholz and K. 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. |
[26] |
M. A. Olshanskii and L. G. Rebholz,
Velocity-vorticity-helicity formulation and a solver for the {N}avier-{S}tokes equations, Journal of Computational Physics, 229 (2010), 4291-4303.
doi: 10.1016/j.jcp.2010.02.012. |
[27] |
A. Palha and M. Gerritsma,
A mass, energy, enstrophy and vorticity conserving (MEEVC) mimetic spectral element discretization for the 2D incompressible Navier-Stokes equations, Journal of Computational Physics, 328 (2017), 200-220.
doi: 10.1016/j.jcp.2016.10.009. |
[28] |
R. Temam,
Navier-Stokes Equations: Theory and Numerical Analysis, North Holland Publishing Company, New York, 1977. |
[29] |
K. L. Wong and A. J. Baker,
A 3D incompressible Navier-Stokes velocity-vorticity weak form finite element algorithm, Int. J. Numer. Meth. Fluids, 38 (2002), 99-123.
doi: 10.1002/fld.204. |
[30] |
X. H. Wu, J. Z. Wu and J. M. Wu,
Effective vorticity-velocity formulations for the three-dimensional incompressible viscous flows, J. Comput. Phys., 122 (1995), 68-82.
doi: 10.1006/jcph.1995.1197. |
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