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Inf-sup stable nonconforming finite elements of arbitrary order on triangles. (English) Zbl 1089.65123
The main objective of this paper is to propose a consistent family of nonconforming approximations of arbitrary order $$k$$ and to prove the inf-sup stability of its vector-valued version with discontinuous piecewise $$P_{k-1}$$ approximations. A general convergence result for nonconforming discretisations of the stationary Stokes problem is presented with a new family of scalar nonconforming finite elements of order $$k - 1$$. Also the basic property of unisolvence of the set of degrees of freedom is shown. The theoretical convergence results are confirmed by numerical experiments.

##### MSC:
 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 35J25 Boundary value problems for second-order elliptic equations 65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
MooNMD
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##### References:
 [1] Arnold, D.N., Brezzi, F.: Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates. RAIRO Modél. Math. Anal. Numér. 19(1), 7–32 (1985) · Zbl 0567.65078 [2] Bercovier, M., Pironneau, O.: Error estimates for finite element solution of the Stokes problem in primitive variables. Numer. Math. 33, 211–224 (1979) · Zbl 0423.65058 · doi:10.1007/BF01399555 [3] Bernardi, C., Raugel, G.: Analysis of some finite elements for the Stokes problem. Math. Comput. 44, 71–79 (1985) · Zbl 0563.65075 · doi:10.1090/S0025-5718-1985-0771031-7 [4] Braess, D., Sarazin, R.: An efficient smoother for the Stokes problem. Applied Numer. Math. 23(1), 3–19 (1997) · Zbl 0874.65095 · doi:10.1016/S0168-9274(96)00059-1 [5] Brezzi, F., Falk, R.S.: Stability of higher-order Taylor-Hood methods. SIAM J. Numer. Anal. 28(3), 581–590 (1991) · Zbl 0731.76042 · doi:10.1137/0728032 [6] Ciarlet, P.G.: The finite element method for elliptic problems. North-Holland Publishing Company, Amsterdam, New York Oxford, 1978 · Zbl 0383.65058 [7] Crouzeix, M., Falk, R.S.: Nonconforming finite elements for the Stokes problem. Math. Comput. 52(186), 437–456 (1989) · Zbl 0685.76018 [8] Crouzeix, M., Raviart, P.-A.: Conforming and nonconforming finite element methods for solving the stationary Stokes equations I. RAIRO Anal. Numér. 7, 33–76 (1973) · Zbl 0302.65087 [9] Fortin, M.: An analysis of the convergence of mixed finite element methods. RAIRO Anal. Numér. 11, 341–354 (1977) · Zbl 0373.65055 [10] Fortin, M., Soulie, M.: A non-conforming piecewise quadratic finite element on triangles. Int. J. Num. Meth. Eng. 19, 502–520 (1983) · Zbl 0514.73068 · doi:10.1002/nme.1620190405 [11] Girault, V., Raviart, P.-A.: Finite Element Methods for Navier–Stokes equations. Springer-Verlag, Berlin-Heidelberg New York, 1986 · Zbl 0585.65077 [12] John, V.: Large Eddy Simulation of Turbulent Incompressible Flows. Analytical and Numerical Results for a Class of LES Models, vol 34: Lecture Notes in Computational Science and Engineering. Springer-Verlag Berlin, Heidelberg New York, 2003 [13] John, V., Knobloch, P., Matthies, G., Tobiska, L.: Non-nested Multi-level Solvers for Finite Element Discretisations of Mixed Problems. Computing 68(4), 313–341 (2002) · Zbl 1006.65137 · doi:10.1007/s00607-002-1444-2 [14] John, V., Matthies, G.: MooNMD - program package based on mapped finite element methods. Comput. Visual. Sci. 6, 163–170 (2004) · Zbl 1061.65124 [15] Knobloch, P.: On the application of the P1mod element to incompressible flow problems. Comput. Visual. Sci. 6, 185–195 (2004) · doi:10.1007/s00791-004-0127-2 [16] Knobloch, P., Tobiska, L.: Modified FE discretizations of incompressible flow problems and their relationship to stabilized methods. In: Neittaanmäki, P., Tiihonen, T., Tarvainen, P. (eds.) Proceedings of the 3rd European Conference Numerical Mathematics and Advanced Applications,. World Scientific, Singapore, 2000 pp. 571–578 · Zbl 1006.76057 [17] Mansfield, L.: Finite element subspaces with optimal rates of convergence for the stationary Stokes problem. RAIRO Anal. Numér. 16(1), 49–66 (1982) · Zbl 0477.65084 [18] Maubach, J.M., Rabier, P.J.: Nonconforming finite elements of arbitrary degree over triangles. RANA report 0328, Technical University of Eindhoven, 2003 [19] Schieweck, F.: A general transfer operator for arbitrary finite element spaces. Preprint 00-25, Fakultät für Mathematik, Otto-von-Guericke-Universität Magdeburg, 2000 [20] Scott, L.R., Vogelius, M.: Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials. RAIRO Anal. Numér. 19, 111–143 (1985) · Zbl 0608.65013 [21] Vanka, S.: Block-implicit multigrid calculation of two-dimensional recirculating flows. Comp. Meth. Appl. Mech. Eng. 59(1), 29–48 (1986) · Zbl 0604.76025 · doi:10.1016/0045-7825(86)90022-8 [22] Verfürth, R.: Error estimates for a mixed finite element approximation of the Stokes equations. RAIRO Anal. Numér. 18, 175–182 (1984) · Zbl 0557.76037
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