Divergence stability in connection with the p-version of the finite element method. (English) Zbl 0717.65085

The paper analyzes certain stability properties for approximations of elliptic problems with divergence constraints. As an important example two-dimensional Stokes equations with appropriate boundary conditions are considered. The discretization is done by replacing velocity and pressure spaces of the standard weak formulation by finite dimensional subspaces. The main task is to find velocity spaces \(V_ N\) and pressure spaces \(W_ N\) so that the discretization is stable and at the same time has good approximation properties. If \(W_ N\) is choosen equal to div \(V_ N\) then the requirement that the divergence operator \(V_ N\to W_ N\) has uniformly bounded right inverses is sufficient for stability and quasi-optimal error estimates. For piecewise polynomial velocities of fixed degree p (p\(\geq 4)\) the divergence operator possesses maximal right inverses bounded independently of the mesh size h.
In this paper the authors demonstrate with a few examples, theoretical as well as computational, that it is not in general possible to find maximal right inverses for the divergence operator, acting on entire polynomials, the norms of which are bounded uniformly in p. They discuss both spaces of total and separate degree \(\leq p\), as well as spaces with and without boundary conditions.
Reviewer: J.Weisel


65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
76M10 Finite element methods applied to problems in fluid mechanics
76D07 Stokes and related (Oseen, etc.) flows
35J25 Boundary value problems for second-order elliptic equations
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[1] [1] D. N. ARNOLD, L. R. SCOTT and M. VOGELIUS, Regular inversion of the divergence operator with Dirichlet boundary conditions on a polygon. Annali Scuola Norm. Sup. Pisa, Serie 4, 15 (1988), pp. 169-192. Zbl0702.35208 MR1007396 · Zbl 0702.35208
[2] [2] I. BABUŠKA, Error-bounds for the finite element method. Numerische Mathematik, 16 (1971), pp. 322-333. Zbl0214.42001 MR288971 · Zbl 0214.42001 · doi:10.1007/BF02165003
[3] I. BABUŠKA, B. A. SZABO and I. N. KATZ, The p-version of the finite element method. SIAM J. Numer. Anal., 18 (1981), pp. 515-545. Zbl0487.65059 MR615529 · Zbl 0487.65059 · doi:10.1137/0718033
[4] [4] M. BERCOVIER and O. PIRONNEAU, Error estimates for the finite element method solution of the Stokes problem in the primitive variables. Numerische Mathematik, 33 (1979), pp. 211-224. Zbl0423.65058 MR549450 · Zbl 0423.65058 · doi:10.1007/BF01399555
[5] [5] J. BOLAND and R. A. NICOLAIDES, On the stability of Bilinear-Constant Velocity-Pressure Finite Elements. Numerische Mathematik, 44 (1984), pp. 219-222. Zbl0544.76030 MR753954 · Zbl 0544.76030 · doi:10.1007/BF01410106
[6] J. BOLAND and R. A. NICOLAIDES, Stable and semistable low order finite elements for viscous flows. SIAM J. Numer. Anal., 22 (1985), pp. 474-492. Zbl0578.65123 MR787571 · Zbl 0578.65123 · doi:10.1137/0722028
[7] [7] F. BREZZI, On the existence, uniqueness and approximation of saddlepoint problems arising from Lagrangian multipliers, RAIRO 8 (1974), pp. 129-151. Zbl0338.90047 MR365287 · Zbl 0338.90047
[8] [8] C. CANUTO, Y. MADAY and A. QUARTERONI, Combined Finite Element and Spectral approximation of the Navier-Stokes equations. Numerische Mathematik, 44 (1984), pp. 201-217. Zbl0614.76021 MR753953 · Zbl 0614.76021 · doi:10.1007/BF01410105
[9] V. GIRAULT and P.-A. RAVIART, Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms. Springer-Verlag, 1986. Zbl0585.65077 MR851383 · Zbl 0585.65077
[10] D. GOTTLIEB and S. ORSZAG, Numerical Analysis of Spectral Methods : Theory and Applications. CBMS-NSF Regional Conference Series in Applied Mathematics, 26. SIAM, 1977. Zbl0412.65058 MR520152 · Zbl 0412.65058
[11] P. GRISVARD, Elliptic Problems in Nonsmooth Domains. Pitman, 1985. Zbl0695.35060 MR775683 · Zbl 0695.35060
[12] C. JOHNSON and J. PITKÄRANTA, Analysis of some mixed finite element methods related to reduced integration. Math, of Comp., 38 (1982), pp. 375-400. Zbl0482.65058 MR645657 · Zbl 0482.65058 · doi:10.2307/2007276
[13] R. B. KELLOGG and J. E. OSBORN, A regularity result for the Stokes problem in a convex polygon. J. Functional Analysis, 21 (1976), pp. 397-431. Zbl0317.35037 MR404849 · Zbl 0317.35037 · doi:10.1016/0022-1236(76)90035-5
[14] N. N. LEBEDEV, Special Functions and Their Applications, Prentice-Hall, 1965. Zbl0131.07002 MR174795 · Zbl 0131.07002
[15] Y. MADAY, Analysis of spectral projectors in multi-dimensional domains. · Zbl 0745.41033 · doi:10.2307/2008432
[16] E. M. RONQUIST, Optimal spectral element method for the unsteady three-dimensional incompressible Navier-Stokes equations. Ph. D. thesis, M.I.T., 1988.
[17] [17] G. SACCHI LANDRIANI, Spectral tau approximation of the two-dimensional Stokes problem. Numer. Math., 52 (1988), pp. 683-699. Zbl0629.76037 MR946383 · Zbl 0629.76037 · doi:10.1007/BF01395818
[18] [18] L. R. SCOTT and M. VOGELIUS, Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials. Math. Modelling Num. Anal., 19 (1985), pp. 111-143. Zbl0608.65013 MR813691 · Zbl 0608.65013
[19] L. R. SCOTT and M. VOGELIUS, Conforming Finite Element Methods for incompressible and nearly incompressible continua. Lectures in Applied Mathematics, 22, pp. 221-244, AMS, 1985. Zbl0582.76028 MR818790 · Zbl 0582.76028
[20] M. SURI, On the stability and convergence of higher order mixed finite element methods for second order elliptic problems. Math. Comput., 54 (1990), pp. 1-19. Zbl0687.65101 MR990603 · Zbl 0687.65101 · doi:10.2307/2008679
[21] B. A. SZABO, I. BABUŠKA and B. K. CHAYAPATHY, Stress computations for nearly incompressible materials. Int. J. Numer. Methods Eng., 28 (1990), pp. 2175-2190. Zbl0718.73083 · Zbl 0718.73083 · doi:10.1002/nme.1620280913
[22] [22] R. VERFÜHRT, Error estimates for a mixed finite element approximation of the Stokes equations. RAIRO, Numer. Anal., 18 (1984), pp 175-182. Zbl0557.76037 MR743884 · Zbl 0557.76037
[23] [23] M. VOGELIUS, An analysis of the p-Version of the Finite Element Method for nearly incompressible materials. Uniformly valid, optimal error estimates. Numerische Mathematik, 41 (1983), pp. 39-53. Zbl0504.65061 MR696549 · Zbl 0504.65061 · doi:10.1007/BF01396304
[24] [24] M. VOGELIUS, A right inverse for the divergence operator in spaces of piecewise polynomials. Application to the p-Version of the Finite Element Method. Numerische Mathematik, 41 (1983), pp. 19-37. Zbl0504.65060 MR696548 · Zbl 0504.65060 · doi:10.1007/BF01396303
[25] [25] C. BERNARDI, Y. MADAY and B. METIVET, Spectral approximation of the periodic-nonperiodic Navier-Stokes equations. Numerische Mathematik, 51 (1987), pp. 655-700. Zbl0583.65085 MR914344 · Zbl 0583.65085 · doi:10.1007/BF01400175
[26] C. BERNARDI, Y. MADAY and B. METIVET, Calcul de la pression dans la résolution spectrale du problème de Stokes. La Recherche Aérospatiale, 1 (1987), pp. 1-21. Zbl0642.76037 MR904608 · Zbl 0642.76037
[27] [27] M. SURI, The p-version of the finite element method for elliptic equations of order 2 l. M2AN, 24 (1990), pp. 265-304. Zbl0711.65094 MR1052150 · Zbl 0711.65094
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