A family of C 1 finite elements with optimal approximation properties for various Galerkin methods for 2nd and 4th order problems
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 13 (1979) no. 3, p. 227-255
@article{M2AN_1979__13_3_227_0,
     author = {Douglas, Jim Jr. and Dupont, Todd and Percell, Peter and Scott, Ridgway},
     title = {A family of $C^1$ finite elements with optimal approximation properties for various Galerkin methods for 2nd and 4th order problems},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {Dunod},
     volume = {13},
     number = {3},
     year = {1979},
     pages = {227-255},
     zbl = {0419.65068},
     mrnumber = {543934},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_1979__13_3_227_0}
}
Douglas, Jim Jr.; Dupont, Todd; Percell, Peter; Scott, Ridgway. A family of $C^1$ finite elements with optimal approximation properties for various Galerkin methods for 2nd and 4th order problems. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 13 (1979) no. 3, pp. 227-255. http://www.numdam.org/item/M2AN_1979__13_3_227_0/

1. S. Gmon, Elliptic Boundary Values Problems, D. van Nostrand, 1965.

2. A. Berger, R. Scott and G. Strang, Approximate Boundary Conditions in the Finite Element Method, Symposia Mathematica, X, Academic Press, 1972, pp. 295-313. | MR 403258 | Zbl 0266.73050

3. S. Bergman and M. Schiffer, Kernel Functions and Elliptic Differential Equations in Mathematical Physics, Academic Press, 1953. | MR 54140 | Zbl 0053.39003

4. J. J. Blair, Higher Order Approximations to the Boundary Conditions for the Finite element Method, Math. Comp., Vol. 30, 1976, pp. 250-262. | MR 398123 | Zbl 0342.65068

5. J. H. Bramble and S. R. Hilbert, Estimation of Linear Functionals on Sobolev Spaces with Applications to Fourier Transforms and Spline Interpolation, S.I.A.M. J. Numer. Anal., Vol. 7, 1970, pp. 112-124. | MR 263214 | Zbl 0201.07803

6. J. H. Bramble and A. H. Schatz, Rayleigh-Ritz-Galerkin Methods for Dirichlet's Problem Using Subspaces Without Boundary Conditions, Comm. Pure App. Math.,Vol. 23, 1970, pp. 653-674. | MR 267788 | Zbl 0204.11102

7. J. H. Bramble and A. H. Schatz, Least Squares Methods for 2m-th Order Elliptic Boundary-Value Problems, Math. Comp., Vol. 25, 1971, pp. 1-32. | MR 295591 | Zbl 0216.49202

8. P. G. Ciarlet, Sur l'élément de Clough et Tocher, R.A.I.R.O., Analyse numérique, Vol. 2, 1974, pp. 19-27. | Numdam | MR 381349 | Zbl 0306.65070

9. P. G. Ciarlet, Numerical Analysis of the Finite Element Method, Séminaire de Mathématiques supérieures, Université de Montréal, 1975. | MR 495010 | Zbl 0363.65083

10. P. G. Ciarlet and P.-A. Raviart, General Lagrange and Hermite Interpolation in Rn with Applications to Finite Element Methods, Arch. Rational Mech. Anal., Vol. 46, 1972, pp. 177-199. | MR 336957 | Zbl 0243.41004

11. J. F. Ciavaldini and J. C. Nedelec, Sur l'élement de Fraeijs de Veubeke et Sander, R.A.I.R.O., Analyse numérique, Vol. 2, 1974, pp. 29-46. | Numdam | MR 381350 | Zbl 0304.65077

12. R. W. Clough and J. L. Tocher, Finite Element Stiffness Matrices and Analysis of Plates in Bending, Proceedings of Conference on Matrix Methods in Structural Mechanics, Wright-Patterson AFB, 1965.

13. J. Jr. Douglas, H1-Galerkin Methods for a Nonlinear Dirichlet Problem, Mathematical Aspects of Finite Element Methods, Rome, 1975, Lecture Notes in Mathematics, n° 606, Springer-Verlag, 1977, pp. 64-86. | MR 471369 | Zbl 0371.65023

14. J. Jr. Douglas and T. Dupont, Collocation Methods for Parabolic Equations in a Single Space Variable, Lecture Notes in Mathematics, n° 385, Springler-Verlag, 1974. | MR 483559 | Zbl 0279.65097

15. J. Jr. Douglas and T. Dupont, H~1-Galerkin Methods for Problems Involving Several space Variables, Topics in Numerical Analysis, III, John J. H. MILLER, éd., Academic Press, 1977, pp. 125-141. | MR 657982 | Zbl 0436.65084

16. J. Jr. Douglas, T. Dupont and M. F. Wheeler, H1-Galerkin Methods for the Laplace and Heat Equations, Mathematical Aspects of Finite Elements in Partial Differential Equations, C. DEBOOR, éd., Academic Press, 1974, pp. 383-416. | MR 349031 | Zbl 0347.65047

17. T. Dupont and R. Scott, Polynomial Approximation of Functions in Sobolev Spaces, submitted Math. Comp. | MR 559195 | Zbl 0423.65009

18. B. Fraeijs Deveubeke, Bending and Stretching of Plates, Proceedings of Conference on Matrix Methods in Structural Mechanics, Wright-Patterson AFB, 1965.

19. P. Grisvard, Behavior of the Solutions of an Elliptic Boundary Value Problem in Polygonal or Polyhedral Domain, Numerical Solution of Partial Differential Equations, III (Synspade, 1975), Bert HUBBARD, éd., Academic Press, 1976,pp.207-274. | MR 466912 | Zbl 0361.35022

20. P. Jamet, Estimation de Verreur d'interpolation dans un domaine variable et application aux éléments finis quadrilatéraux dégénérés, in Méthodes numériques enmathématiques appliquées (Séminaire de Mathématiques supérieures, été 1975),Presses de l'Université de Montréal, Vol. 60, 1977. | Zbl 0374.65007

21. I. N. Katz, A. G. Peano and B. A. Szabo, Nodal Variables for Arbitrary Order Conforming Finite Eléments, U. S. Dept. of Transportation Tech. Rep. DOT-OS-30108-5, Washington Univ., June, 1975.

22. J. L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, Vol. 1, Dunod, Paris, 1968. | MR 247243 | Zbl 0165.10801

23. J. Morgan and R. Scott, A nodal basis for C 1 piecewise polynomials of degree n5, Math. Comp., Vol. 29, 1975, pp. 736-740. | MR 375740 | Zbl 0307.65074

24. J. Nitsche, On Dirichlet Problems Usina Subspaces with Nearly Zero Boundary Conditions, The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, A.K. Aziz, éd., Academic Press, 1972, pp. 603-628. | MR 426456 | Zbl 0271.65059

25. A.G. Peano, Hierarchies of Conforming Finite Elements for Plane Elasticity and Plate Bending, Comp. and Maths, with Appls., VoL 2, 1976t pp. 211-224. | Zbl 0369.73071

26. P. Percell, On Cubic and Quartic Clough-Tocher Finite Eléments, S.I.A.M.J. Numer. Anal., Vol. 13, 1976, pp. 100-103. | MR 408198 | Zbl 0319.65064

27. G. Sander, Bornes supérieures et inférieures dans l'analyse matricielle des plaques en flexion-torsion, Bull. Soc. Royale des Se. de Liège, Vol. 33, 1964, pp, 456-494. | MR 170526

28. A. H. Schatz, An Observation Concerning Ritz-Galerkin Methods with Indefinite Bilinear Forms, Math. Comp., Vol. 28, 1974, pp. 959-962. | MR 373326 | Zbl 0321.65059

29. R. Scott, C1 Continuity via Constraints for 4th Order Problems, Mathematical Aspects of Finite Eléments in Partial Differential Equations, C. DEBOOR, éd., Academic Press; 1974, pp. 171-193. | MR 658318 | Zbl 0337.65059

30. R. Scott, Interpolatetl Boundarv Conditions in the Finite Element Method, S.LA.M.J. Numer. Anal., Vol. 12, 1975, pp. 404-427. | MR 386304 | Zbl 0357.65082

31. G. Strang, Piecewise Polynomials and the Finite Element Method, Bull. A.M.S.,VoL 79, 1973, pp. 1128-1137. | MR 327060 | Zbl 0285.41009

32. B.A. Szabo et al., Advanced Design Technology for Rail Transportation Vehicles, .S.Dept. of Transportation Tech. Rep. DOT-OS-30108-2, Washington Univ., June, 1974.

33. V. Thomée and L. Wahlbin, On Galerkin Methods in Semi-Linear Parabolie Problems, S.I.A.M. J. Num. Anal., VoL 12, 1975, pp. 378-389. | MR 395269 | Zbl 0307.35007

34. O. C. Zienkiewicz, TheFinite Element Method in Engineering Science, McGraw-Hill, 1971. | MR 315970 | Zbl 0237.73071