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A new finite element formulation for computational fluid dynamics. VII. The Stokes problem with various well-posed boundary conditions: Symmetric formulations that converge for all velocity/pressure spaces. (English) Zbl 0635.76067
[For part VI see the review above (Zbl 0635.76066).]
Symmetric finite element formulations are proposed for the primitive- variables form of the Stokes equations and shown to be convergent for any combination of pressure and velocity interpolations. Various boundary conditions, such as pressure, are accomodated.

76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
80A20 Heat and mass transfer, heat flow (MSC2010)
65Z05 Applications to the sciences
Full Text: DOI
[1] Arnold, D., An interior penalty finite element method with discontinuous elements, SIAM J. numer. anal., 19, 742-760, (1982) · Zbl 0482.65060
[2] Babuška, I., Error bounds for finite element method, Numer. math., 16, 322-333, (1971) · Zbl 0214.42001
[3] Brezzi, F., On the existence, uniqueness and approximation of saddle-point problems arising from Lagrange multipliers, RAIRO ser rouge anal. numér., R-2, 129-151, (1979) · Zbl 0338.90047
[4] Carey, G.F.; Oden, J.T., Finite elements: A second course II, (1983), Prentice-Hall Englewood Cliffs, NJ · Zbl 0515.65075
[5] Franca, L.P., New mixed finite element methods, () · Zbl 0651.65078
[6] Franca, L.P.; Hughes, T.J.R.; Loula, A.F.D.; Miranda, I., A new family of stable elements for nearly incompressible elasticity based on a mixed Petrov-Galerkin finite element method, (), also Numer. Math. (to appear).
[7] Girault, V.; Raviart, P.A., Finite element methods for Navier-Stokes equations, () · Zbl 0396.65070
[8] Hellinger, E., Der allgemeine ansatz der mechanik der kontinua, (), 602-694, (4)
[9] Herrmann, L.R., Elasticity equations for nearly incompressible materials by a variational theorem, Aiaa j., 3, 1896-1900, (1965)
[10] Hughes, T.J.R., The finite element method: linear static and dynamic finite element analysis, (1987), Prentice-Hall Englewood Cliffs, NJ
[11] Hughes, T.J.R.; Franca, L.P.; Balestra, M., 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, Comput. meths. appl. mech. engrg., 59, 85-99, (1986) · Zbl 0622.76077
[12] Jaunzemis, W., Continuum mechanics, (1967), MacMillan New York · Zbl 0173.52103
[13] Loula, A.F.D.; Franca, L.P.; Hughes, T.J.R.; Miranda, I., Stability, convergence and accuracy of a new finite element method for the circular arch problem, Comput. meths. appl. mech. engrg., 63, 281-303, (1987) · Zbl 0607.73077
[14] Loula, A.F.D.; Hughes, T.J.R.; Franca, L.P.; Miranda, I., Mixed Petrov-Galerkin method for the Timoshenko beam, Comput. meths. appl. mech. engrg., 63, 133-154, (1987) · Zbl 0607.73076
[15] Loula, A.F.D.; Miranda, I.; Hughes, T.J.R.; Franca, L.P., A successful mixed formulation for axisymmetric shell analysis employing discontinuous stress fields of the same order as the displacement field, ()
[16] Pironneau, O., Conditions aux limites sur la pression pour LES équations de Stokes et de Navier-Stokes, C.R. acad. sc. Paris, t. 303, 9, 403-406, (1986), Série I · Zbl 0613.76028
[17] Reissner, E., On a variational theorem in elasticity, J. math. phys., 29, 2, 90-95, (1950) · Zbl 0039.40502
[18] Thomasset, F., Implementation of finite element methods for Navier-Stokes equations, (1981), Springer New York · Zbl 0475.76036
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