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A new finite element formulation for computational fluid dynamics. V: Circumventing the Babuška-Brezzi condition: A stable Petrov-Galerkin formulation of the Stokes problem accommodating equal-order interpolations. (English) Zbl 0622.76077
[For the former parts see the above entries.]
A new Petrov-Galerkin formulation of the Stokes problem is proposed. The new formulation possesses better stability properties than the classical Galerkin/variational method. An error analysis is performed for the case in which both the velocity and pressure are approximated by \(C^ 0\) interpolations. Combinations of \(C^ 0\) interpolations which are unstable according to the Babuška-Brezzi condition (e.g., equal-order interpolations) are shown to be stable and convergent within the present framework. Calculations exhibiting the good behavior of the methodology are presented.

MSC:
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
65Z05 Applications to the sciences
76N15 Gas dynamics (general theory)
76R99 Diffusion and convection
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[1] Babuška, I., Error bounds for finite element method, Numer. math., 16, 322-333, (1971) · Zbl 0214.42001
[2] Brezzi, F., On the existence, uniqueness and approximation of saddle-point problems arising from Lagrange multipliers, Rev. française d’automatique inform. rech. opér., ser. rouge anal. numér., 8, R-2, 129-151, (1974) · Zbl 0338.90047
[3] Brezzi, F.; Pitkäranta, J., On the stabilization of finite element approximations of the Stokes equations, (), 11-19, also in · Zbl 0552.76002
[4] Hellinger, E., Der allgemeinen ansätze der mechanik der kontinua, (), 602-694, Part 4 · JFM 45.1012.01
[5] Herrmann, L.R, Elasticity equations for nearly incompressible elasticity, Aiaa j., 3, 1896-1900, (1965)
[6] Hughes, T.J.R.; Brooks, A.N., A theoretical framework for Petrov-Galerkin methods with discontinuous weighting functions: application to the streamline upwind procedure, (), 46-65
[7] Johnson, C., Streamline diffusion methods for problems in fluid mechanics, (), 251-261
[8] Johnson, C.; Saranen, J., Streamline diffusion methods for the incompressible Euler and Navier-Stokes equations, (1984), preprint
[9] Oden, J.T.; Carey, G.F., ()
[10] Reissner, E., On a variational theorem in elasticity, J. math. phys., 29, 2, 90-95, (1950) · Zbl 0039.40502
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