# On a 2D vector Poisson problem with apparently mutually exclusive scalar boundary conditions

• Volume: 34, Issue: 1, page 183-200
• ISSN: 0764-583X

top

## Abstract

top
This work is devoted to the study of a two-dimensional vector Poisson equation with the normal component of the unknown and the value of the divergence of the unknown prescribed simultaneously on the entire boundary. These two scalar boundary conditions appear prima facie alternative in a standard variational framework. An original variational formulation of this boundary value problem is proposed here. Furthermore, an uncoupled solution algorithm is introduced together with its finite element approximation. The numerical scheme has been implemented and applied to solve a simple test problem.

## How to cite

top

Guermond, Jean-Luc, Quartapelle, Luigi, and Zhu, Jiang. "On a 2D vector Poisson problem with apparently mutually exclusive scalar boundary conditions." ESAIM: Mathematical Modelling and Numerical Analysis 34.1 (2010): 183-200. <http://eudml.org/doc/197483>.

@article{Guermond2010,
abstract = { This work is devoted to the study of a two-dimensional vector Poisson equation with the normal component of the unknown and the value of the divergence of the unknown prescribed simultaneously on the entire boundary. These two scalar boundary conditions appear prima facie alternative in a standard variational framework. An original variational formulation of this boundary value problem is proposed here. Furthermore, an uncoupled solution algorithm is introduced together with its finite element approximation. The numerical scheme has been implemented and applied to solve a simple test problem. },
author = {Guermond, Jean-Luc, Quartapelle, Luigi, Zhu, Jiang},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Vector Poisson equation; mutually exclusive scalar boundary conditions; variational formulation; uncoupled solution; finite element approximation.; numerical examples; vector Poisson equation; variational formulations; splitting method; algorithm; finite element methods; error estimates},
language = {eng},
month = {3},
number = {1},
pages = {183-200},
publisher = {EDP Sciences},
title = {On a 2D vector Poisson problem with apparently mutually exclusive scalar boundary conditions},
url = {http://eudml.org/doc/197483},
volume = {34},
year = {2010},
}

TY - JOUR
AU - Guermond, Jean-Luc
AU - Quartapelle, Luigi
AU - Zhu, Jiang
TI - On a 2D vector Poisson problem with apparently mutually exclusive scalar boundary conditions
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 34
IS - 1
SP - 183
EP - 200
AB - This work is devoted to the study of a two-dimensional vector Poisson equation with the normal component of the unknown and the value of the divergence of the unknown prescribed simultaneously on the entire boundary. These two scalar boundary conditions appear prima facie alternative in a standard variational framework. An original variational formulation of this boundary value problem is proposed here. Furthermore, an uncoupled solution algorithm is introduced together with its finite element approximation. The numerical scheme has been implemented and applied to solve a simple test problem.
LA - eng
KW - Vector Poisson equation; mutually exclusive scalar boundary conditions; variational formulation; uncoupled solution; finite element approximation.; numerical examples; vector Poisson equation; variational formulations; splitting method; algorithm; finite element methods; error estimates
UR - http://eudml.org/doc/197483
ER -

## References

top
1. Y. Achdou, R. Glowinski and O. Pironneau, Tuning the mesh of a mixed method for the stream function-vorticity formulation of the Navier-Stokes equations. Numer. Math.63 (1992) 145-163.  Zbl0760.76041
2. I. Babuska, The finite element method with Lagrange multipliers. Numer. Math.20 (1973) 179-192.  Zbl0258.65108
3. C. Bernardi, Méthodes d'éléments finis mixtes pour les équations de Navier-Stokes. Thèse de 3e Cycle, Université de Paris VI, France (1979).
4. P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978).  Zbl0383.65058
5. F. El Dabaghi and O. Pironneau, Stream vectors in three dimensional aerodynamics. Numer. Math.48 (1986) 561-589.  Zbl0625.76009
6. V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations. Springer-Verlag, Berlin (1986).  Zbl0413.65081
7. R. Glowinski and O. Pironneau, Numerical methods for the first biharmonic equation and for the two-dimensional Stokes problem. SIAM Rev.21 (1979) 167-212.  Zbl0427.65073
8. P. Neittaanmaki and M. Krizek, in Efficient Solution of Elliptic Systems, Finite element approximation for a div-rot system with mixed boundary conditions in non-smooth plane domain. Notes in Numerical Fluid Mechanics, Vol. 10, W. Hachbush Ed., Vieweg Publishing, Wiesbaden, Germany (1984); see also Appl. Math.29 (1984) 272-285.  Zbl0575.65125
9. L. Quartapelle, Numerical Solution of the Incompressible Navier-Stokes Equations. Birkhäuser, Basel (1993).  Zbl0784.76020
10. L. Quartapelle and A. Muzzio, Decoupled solution of vector Poisson equations with boundary condition coupling, in Computional Fluid Dynamics, G. de Vahl Davis and C. Fletcher Eds., Elsevier Science Publishers B.V., North-Holland (1988) 609-619.
11. L. Quartapelle, V. Ruas and J. Zhu, Uncoupled solution of the three-dimensional vorticity-velocity equations. ZAMP49 (1998) 384-400.  Zbl0912.35124
12. V. Ruas, L. Quartapelle and J. Zhu, A symmetrized velocity-vorticity formulation of the three-dimensional Stokes system. C.R. Acad. Sci. Paris Sér. IIb323 (1996) 819-824.  Zbl0923.76033
13. G. Strang and G.J. Fix, An Analysis of the Finite Element Method, Prentice-Hall, Englewood Cliffs, New York (1973).  Zbl0356.65096
14. J. Zhu, A.F.D. Loula and L. Quartapelle, A vector Poisson problem with coupling boundary conditions in a Lipschitz 2D domain, Research Report, Laboratório Nacional de Computaç ao Científica, CNPq, N0 30 (1997).
15. J. Zhu, A. F. D. Loula and L. Quartapelle, Finite element solution of vector Poisson equation with a coupling boundary condition. Numer. Methods Partial Differential Eq.16 (2000).  Zbl0956.65103
16. J. Zhu, L. Quartapelle and A.F.D. Loula, Uncoupled variational formulation of a vector Poisson problem. C.R. Acad. Sci. Paris Sér. I 323 (1996) 971-976.  Zbl0869.76069

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.