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On a 2D vector Poisson problem with apparently mutually exclusive scalar boundary conditions. (English) Zbl 0949.65116
This paper considers the two-dimensional vector Poisson equation with two boundary conditions imposing given values to the normal component and to the divergence; these conditions are not natural with respect to the usual variational formulations. The kernels of the associated and transposed operators are studied and the variational formulation is modified and reformulated in a well-posed manner. Then, a splitting method leads to an uncoupled numerical algorithm requiring to solve only scalar Poisson equations and an auxiliary problem for a scalar boundary unknown. Finally, the problem is approximated by finite element methods, corresponding error estimates are obtained and a few numerical tests illustrate the pertinency of the method.

##### MSC:
 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 65N15 Error bounds for boundary value problems involving PDEs 35J50 Variational methods for elliptic systems
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##### References:
 [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 · Zbl 0760.76041 · doi:10.1007/BF01385852 · eudml:133673 [2] I. Babuska, The finite element method with Lagrange multipliers. Numer. Math.20 (1973) 179-192. · Zbl 0258.65108 · doi:10.1007/BF01436561 · eudml:132183 [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). · Zbl 0383.65058 [5] F. El Dabaghi and O. Pironneau, Stream vectors in three dimensional aerodynamics. Numer. Math.48 (1986) 561-589. Zbl0625.76009 · Zbl 0625.76009 · doi:10.1007/BF01389451 · eudml:133087 [6] V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations. Springer-Verlag, Berlin (1986). · Zbl 0585.65077 [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. · Zbl 0427.65073 · doi:10.1137/1021028 [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. · eudml:15357 [9] L. Quartapelle, Numerical Solution of the Incompressible Navier-Stokes Equations. Birkhäuser, Basel (1993). · Zbl 0784.76020 [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. · Zbl 0912.35124 · doi:10.1007/s000000050098 [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. · Zbl 0923.76033 [13] G. Strang and G.J. Fix, An Analysis of the Finite Element Method, Prentice-Hall, Englewood Cliffs, New York (1973). · Zbl 0356.65096 [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). · Zbl 0956.65103 · doi:10.1002/(SICI)1098-2426(200001)16:1<71::AID-NUM6>3.0.CO;2-F [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. · Zbl 0869.76069
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