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A linear nonconforming finite element method for nearly incompressible elasticity and Stokes flow. (English) Zbl 1067.74578
Summary: We introduce a new triangular element for nearly incompressible elasticity and incompressible fluid flow. The method consists of conforming linear elements for one of the displacement (or velocity for flows) components and linear nonconforming elements for ther component. The element is proved to give an optimal approximation, and this is also confirmed by several numerical examples.

74S05 Finite element methods applied to problems in solid mechanics
76M10 Finite element methods applied to problems in fluid mechanics
Full Text: DOI
[1] Arnold, D.N.; Falk, R.S., A uniformly accurate finite element method for the Reissner-Mindlin plate, SIAM J. numer. anal., 26, 1276-1290, (1989) · Zbl 0696.73040
[2] Brezzi, F.; Fortin, M., Mixed and hybrid finite element methods, (1991), Springer-Verlag Berlin · Zbl 0788.73002
[3] Ciarlet, P.G., The finite element method for elliptic problems, (1978), North-Holland Amsterdam · Zbl 0445.73043
[4] Crouzeix, M.; Raviant, P.A., Conforming and nonconforming finite element methods for solving the stationary Stokes equations, RAIRO anal. numér., R-3, 33-76, (1973) · Zbl 0302.65087
[5] Falk, R.S., Nonconforming finite element methods for the equations of linear elasticity, Math. comput., 57, 529-550, (1991) · Zbl 0747.73044
[6] Falk, R.S.; Morley, M., Equivalence of finite element methods for problems in elasticity, SIAM J. numer. anal., 27, 1486-1505, (1990) · Zbl 0722.73068
[7] Franca, L.P.; Hughes, T.J.R., Two classes of mixed finite element methods, Comput. methods appl. mech. engrg., 69, 89-129, (1988) · Zbl 0651.65078
[8] Fortin, M., Old and new finite elements for incompressible flows, Int. J. numer. methods fluids, 1, 347-364, (1981) · Zbl 0467.76030
[9] Franca, L.P.; Stenberg, R., Error analysis of some Galerkin least squares methods for the elasticity equations, SIAM J. numer. anal., 28, 1680-1697, (1991) · Zbl 0759.73055
[10] Girault, V.; Raviart, P.A., Finite element methods for Navier-Stokes equations. theory and algorithms, (1986), Springer-Verlag Berlin · Zbl 0396.65070
[11] Hughes, T.J.R., The finite element method. linear static and dynamic analysis, (1987), Prentice-Hall Englewood Cliffs
[12] Hughes, T.J.R.; Franca, L.P., A new finite element formulation for computational fluid dynamics: part VII. the Stokes problem with various well-posed boundary conditions: symmetric formulations that converge for all velocity/pressure spaces, Comput. methods appl. mech. engrg., 65, 85-96, (1987) · Zbl 0635.76067
[13] Muskhelishvili, N.I., Some basic problems of the mathematical theory of elasticity, (1953), Noordhoff Leiden · Zbl 0052.41402
[14] Pitkäranta, J.; Stenberg, R., Error bounds for the approximation of Stokes problem with bilinear/constant elements on irregular quadrilateral meshes, (), 325-334 · Zbl 0598.76036
[15] Rannacher, R.; Turek, S., Simple nonconforming quadrilateral Stokes element, Numer. methods pdes, 8, 97-112, (1992) · Zbl 0742.76051
[16] Stenberg, R., A technique for analysing finite element methods for viscous incompressible flow, Int. J. numer. methods fluids, 11, 935-948, (1990) · Zbl 0704.76017
[17] Szabó, B.; Babuška, I., Finite element analysis, (1991), Wiley New York
[18] Zienkiewicz, O.C.; Taylor, R.L., ()
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