\(p\)-version of mixed finite element methods for Stokes-like problems. (English) Zbl 0778.76052

Summary: We investigate various sample extremum (saddle point and minimization) problems from the point of view of stability of increasing order mixed finite element methods. Problems discussed include the Stokes, Poisson and linear elasticity problems.


76M10 Finite element methods applied to problems in fluid mechanics
76D07 Stokes and related (Oseen, etc.) flows
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
74B05 Classical linear elasticity
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[1] Babǔska, I., Error bounds for the finite element method, Numer. Math., 16, 322-333 (1971) · Zbl 0214.42001
[2] Brezzi, F., On the existence, uniqueness and approximation of saddlepoint problems arising from Lagrangian multipliers, RAIRO, 129-151 (1974) · Zbl 0338.90047
[3] Scott, L. R.; Vogelius, M., Conforming finite element methods for incompressible and nearly incompressible continua, (Lectures in Applied Mathematics, 22 (1985), Amer. Mathematical Soc: Amer. Mathematical Soc Providence, RI), 221-244 · Zbl 0582.76028
[5] Vogelius, M., A right-inverse for the divergence operator in spaces of piecewise polynomials — Application to the \(p\) version of the finite element method, Numer. Math., 41, 19-37 (1983) · Zbl 0504.65060
[6] Bernardi, C.; Maday, Y.; Métivet, B., Computation of the pressure in the spectral approximation of the Stokes problem, Rech. Aérospat., 1, 1-21 (1987) · Zbl 0642.76037
[7] Scott, L. R.; Vogelius, M., Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials, RAIRO Modél. Math. Anal. Numér., 19, 111-143 (1985) · Zbl 0608.65013
[8] Jensen, S.; Vogelius, M., Divergence stability in connection with the \(p\) version of the finite element method, RAIRO Modél. Math. Anal. Numér., 24, 737-764 (1990) · Zbl 0717.65085
[9] Landriani, G. Sacchi, Spectral tau approximation of the two-dimensional Stokes problem, Numer. Math., 52, 683-699 (1988) · Zbl 0629.76037
[10] Hanna, M. S.; Smith, K. T., Some remarks on the Dirichlet problem in piecewise-smooth domains, Comm. Pure Appl. Math., 20, 575-593 (1967) · Zbl 0154.13003
[11] Dauge, M., Elliptic Boundary Value Problems on Corner Domains, (Lecture Notes in Mathematics, 1341 (1988), Springer: Springer Berlin) · Zbl 0668.35001
[12] Girault, V.; Raviart, P.-A., Finite Element Approximation of the Navier-Stokes Equations, (Lecture Notes in Mathematics, 749 (1979), Springer: Springer Berlin) · Zbl 0413.65081
[13] Jensen, S., On computing the pressure by the \(p\) version of the finite element method for Stokes problem, Numer. Math., 59, 581-601 (1991) · Zbl 0741.76038
[14] Suri, M., The \(p\)-version of the finite element method for elliptic problems of order 2l, RAIRO Modél. Math. Anal. Numér., 24, 265-304 (1990) · Zbl 0711.65094
[15] Jensen, S., An \(H_o^{m\) · Zbl 0734.65085
[16] Temam, R., Navier-Stokes Equations (197), North-Holland: North-Holland Amsterdam · Zbl 0572.35083
[17] Ciariet, P. G., The Finite Element Method for Elliptic Problems (1978), North-Holland: North-Holland Amsterdam
[18] Grisvard, P., Elliptic Problems in Nonsmooth Domains (1985), Pitman: Pitman London · Zbl 0695.35060
[19] Jensen, S.; Suri, M., On the \(L_2\) error for the \(p\)-version of the finite element method over polygonal domains, Comput. Methods Appl. Mech. Engrg., 97, 233-243 (1992) · Zbl 0762.65058
[20] Han, W.; Jensen, S., On the sharpness of certain duality estimates of \(H_0^1\)-projections onto subspaces of piecewise polynomials as dependent on the degree (1991), Unpublished manuscript
[21] Bramble, J. H.; Schatz, A. H., Higher order local accuracy by averaging in the finite element method, Math. Comp., 31, 94-111 (1977) · Zbl 0353.65064
[22] Johnson, C.; Pitkäranta, J., Analysis of some mixed finite element methods related to reduced integration, Math. Comp., 38, 357-400 (1982) · Zbl 0482.65058
[23] Nedelec, J. C., Mixed finite elements in \(R^3\), Numer. Math., 35, 315-341 (1980) · Zbl 0419.65069
[24] Raviart, P. A.; Thomas, J. M., A mixed finite element method for 2-nd order elliptic equations, (Galligani, I.; Magenes, E., Mathematical Aspects of Finite Element Methods. Mathematical Aspects of Finite Element Methods, Lecture Notes in Mathematics, 606 (1977), Springer: Springer Berlin), 292-315
[25] Arnold, D. N.; Babuška, I.; Osborn, J., Finite element methods: principles for their selection, Comput. Methods Appl. Mech. Engrg., 45, 57-96 (1984) · Zbl 0513.73081
[26] Johnson, C., Numerical Solutions of Partial Differential Equations by the Finite Element Method (1987), Cambridge Univ. Press: Cambridge Univ. Press Cambridge
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