Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials

L. R. Scott; M. Vogelius

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (1985)

  • Volume: 19, Issue: 1, page 111-143
  • ISSN: 0764-583X

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Scott, L. R., and Vogelius, M.. "Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 19.1 (1985): 111-143. <http://eudml.org/doc/193439>.

@article{Scott1985,
author = {Scott, L. R., Vogelius, M.},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {piecewise polynomials; triangulations; range of the divergence operator; maximal right inverse; measure of singularity},
language = {eng},
number = {1},
pages = {111-143},
publisher = {Dunod},
title = {Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials},
url = {http://eudml.org/doc/193439},
volume = {19},
year = {1985},
}

TY - JOUR
AU - Scott, L. R.
AU - Vogelius, M.
TI - Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 1985
PB - Dunod
VL - 19
IS - 1
SP - 111
EP - 143
LA - eng
KW - piecewise polynomials; triangulations; range of the divergence operator; maximal right inverse; measure of singularity
UR - http://eudml.org/doc/193439
ER -

References

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  2. [2] I. BABUSKA, K. AZIZ, Survey lectures on the mathematical foundations of the finite element method. In The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, A. K. Aziz, editor, Academic Press, 1972. Zbl0268.65052MR347104
  3. [3] J. M. BOLAND, R.A. NICOLAIDES, Stability of finite elements under devergence constraints, SIAM J. Num. Anal. 20 (1983), pp. 722-731. Zbl0521.76027MR708453
  4. [4] M. CROUZEIX, P. A. RAVIART, Conforming and nonconforming finite element methods for solving the stationary Stokes equations, I. R.A.LR.O. Sér. Rouge 7 (1973), pp. 33-75. Zbl0302.65087MR343661
  5. [5] P.C. DUNNE, Reply to comments by B. Irons on his paper « Complete polynomial displacement fields for finite element method », Aero. J. Roy. Aero. Soc.72 (1973) pp. 710-711. 
  6. [6] G. J. FIX, M. D. GUNZBURGER, R. A. NICOLAIDES, On mixed finite element methods for first order elliptic systems. Numer. Math. 37 (1981), pp. 29-48. Zbl0459.65072MR615890
  7. [7] V. GIRAULT, P. A. RAVIART, Finite Element Approximation of the Navier-Stokes Equation. Lecture Notes in Mathematics, 749, Springer-Verlag, 1979. Zbl0413.65081MR548867
  8. [8] P. GRISVARD, Boundary value problems in non-smooth domains, Lecture Notes # 19, University of Maryland, 1980. 
  9. [9] B. MERCIER, A conforming finite element method for two dimensional, incompressible elasticity, Int. J. Num Meths. Eng. 14 (1979), pp. 942-945. Zbl0397.73065MR533310
  10. [10] J. MORGAN R. SCOTT, A nodal basis for C 1 piecewise polynomials of degree n 5 no 5. Math. Comput. 29 (1975), pp. 736-740. Zbl0307.65074MR375740
  11. [11] J. MORGAN R. SCOTT, The dimension of the space of C 1 piecewise polynomials (Preprint). 
  12. [12] L. R. SCOTT, M. VOGELIUS, Conforming finite element methods for incompressible and nearly incompressible continua. Proceedings of the 1983 Summer Seminar on Large-scale Computations in Fluid Mechanics, S. Osher, editor, Lect. Appl. Math. 22, to appear. Zbl0582.76028MR818790
  13. [13] E. STEIN, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, 1970. Zbl0207.13501MR290095
  14. [14] R. STENBERG, Analysis of mixed finite element methods for the Stokes problem : A unified approach. To appear, Math. Comp. Zbl0535.76037MR725982
  15. [15] G. STRANG, Piecewise polynomials and the finite element method, Bull. AMS 79 (1973), pp, 1128-1137. Zbl0285.41009MR327060
  16. [16] B. A. SZABO, P. K. BASU, D. A. DUNAVANT, D. VASILOPOULOS, Adaptive finite element technology in integrated design and analysis, Report WU/CCM-81/1. Washington Univestity, St. Louis. 
  17. [17] R. TEMAM, Navier-Stokes Equations, North-Holland, 1977. Zbl0383.35057MR769654
  18. [18] M. VOGELIUS, 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 (1983), pp. 19-37. Zbl0504.65060MR696548
  19. [19] M. VOGELIUS, An analysis of thep-version of the finite element method for nearly incompressible materials. Uniformly valid, optimal error estimates. Numer. Math. 41 (1983), pp. 39-53. Zbl0504.65061MR696549

Citations in EuDML Documents

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  1. Carolina C. Manica, Monika Neda, Maxim Olshanskii, Leo G. Rebholz, Enabling numerical accuracy of Navier-Stokes-α through deconvolution and enhanced stability
  2. Douglas N. Arnold, L. Ridgway Scott, Michael Vogelius, Regular inversion of the divergence operator with Dirichlet boundary conditions on a polygon
  3. Carolina C. Manica, Monika Neda, Maxim Olshanskii, Leo G. Rebholz, Enabling numerical accuracy of Navier-Stokes- through deconvolution and enhanced stability
  4. Petr Knobloch, On stability of the P n mod / P n element for incompressible flow problems
  5. Richard S. Falk, A Fortin operator for two-dimensional Taylor-Hood elements
  6. P. Peisker, D. Braess, Uniform convergence of mixed interpolated elements for Reissner-Mindlin plates
  7. J. Baranger, D. Sandri, A formulation of Stokes's problem and the linear elasticity equations suggested by the Oldroyd model for viscoelastic flow
  8. Jason S. Howell, Noel J. Walkington, Dual-mixed finite element methods for the Navier-Stokes equations
  9. Yongxing Shen, Adrian J. Lew, A family of discontinuous Galerkin mixed methods for nearly and perfectly incompressible elasticity
  10. Vivette Girault, Larkin Ridgway Scott, Finite-element discretizations of a two-dimensional grade-two fluid model

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