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General Lagrange and Hermite interpolation in \(R^n\) with applications to finite element methods. (English) Zbl 0243.41004


MSC:

41A05 Interpolation in approximation theory
41A63 Multidimensional problems
65L10 Numerical solution of boundary value problems involving ordinary differential equations
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[1] Aubin, J.P., Behavior of the approximate solution of boundary value problems for linear elliptic operators by Galerkin’s and finite difference methods. Ann. Scuola Norm. Sup. Pisa 21, 559–639 (1967).
[2] Aubin, J.P., Approximation des espaces de distributions et des opérateurs différentiels. Bull. Soc. Math. France (1967), Mémoire No 12.
[3] Babuška, Ivo, The finite element method for elliptic differential equations (Symp. on the Numerical Solution of Partial Differential Equations, Univ. of Maryland, 1970). Numerical Solution of Partial Differential Equations-II (B. Hubbard, ed.), p. 69–106. New York: Academic Press 1971.
[4] Babuška, Ivo, Error bounds for finite element method. Numer. Math. 16, 322–333 (1971). · Zbl 0214.42001 · doi:10.1007/BF02165003
[5] Bell, K., A refined triangular plate bending finite element. Int. J. Numer. Math. Engineering 1, 101–121 (1969). · doi:10.1002/nme.1620010108
[6] Birkhoff, G., M. Schultz, & R.S. Varga, Piecewise Hermite interpolation in one and two variables with applications to partial differential equations. Numer. Math. 11, 232–256 (1968). · Zbl 0159.20904 · doi:10.1007/BF02161845
[7] Bramble, James H., Variational Methods for the Numerical Solution of Elliptic Problems (Lecture notes). Chalmers Institute of Technology and the University of Göteborg, Sweden, 1970.
[8] Bramble, J.H., & S.R. Hilbert, Estimation of linear functionals on Sobolev spaces with applications to Fourier transforms and spline interpolation. SIAM J. Numer. Anal. 7, 112–124 (1970). · Zbl 0201.07803 · doi:10.1137/0707006
[9] Bramble, J.H., & S.R. Hilbert, Bounds for a class of linear functionals with applications to Hermite interpolation. Numer. Math. 16, 362–369 (1971). · Zbl 0214.41405 · doi:10.1007/BF02165007
[10] Bramble, James H., & Miloš Zlámal, Triangular elements in the finite element method. Math. Comp. 24, 809–820 (1970). · doi:10.1090/S0025-5718-1970-0282540-0
[11] Cartan, Henri, Calcul Differentiel. Paris: Hermann 1967.
[12] Ciarlet, P.G., Some results in the theory of nonnegative matrices. Linear Algebra and Appl. 1, 139–152 (1968). · Zbl 0167.03101 · doi:10.1016/0024-3795(68)90054-2
[13] Ciarlet, P.G., & C. Wagschal, Multipoint Taylor formulas and applications to the finite element method. Numer. Math. 17, 84–100 (1971). · Zbl 0199.50104 · doi:10.1007/BF01395869
[14] Coatmélec, Christian, Approximation et interpolation des fonctions différentiables de plusieurs variables. Ann. Sci. Ecole Norm. Sup. (3) 83, 271–341 (1966). · Zbl 0155.10902 · doi:10.24033/asens.1157
[15] Dieudonnè, J., Foundations of Modern Analysis. New York: Academic Press Inc. 1960. · Zbl 0100.04201
[16] Fix, G., & G. Strang, Fourier analysis of the finite element method in Ritz-Galerkin theory. Studies in Appl. Math. 48, 265–273 (1969). · Zbl 0179.22501 · doi:10.1002/sapm1969483265
[17] Frederickson, Paul O., Triangular spline interpolation. Math. Report No. 6, Lakehead Univ. 1970.
[18] Guenther, R.B., & E. L. Roetman, Some observations on interpolation in higher dimensions. Math. Comp. 24, 517–522 (1970). · Zbl 0234.65010 · doi:10.1090/S0025-5718-1970-0275631-1
[19] Guglielmo, F. di, Méthode des éléments finis: Une famille d’approximation des espaces de Sobolev par les translatées de p fonctions. Calcolo 7, 185–234 (1970). · Zbl 0216.15801 · doi:10.1007/BF02575560
[20] Hall, C.A., Bicubic interpolation over triangles. J. Math. Mech. 19, 1–11 (1969). · Zbl 0194.47102
[21] Nečas, Jindřich, Les Méthodes Directes en Théorie des Equations Elliptiques. Paris: Masson 1967.
[22] Nicolaides, R.A., On Lagrange interpolation in n variables. Institute of Computer Sciences (London), Tech. Note ICSI 274, 1970.
[23] Nicolaides, R.A., On a class of finite elements generated by Lagrange interpolation II. Institute of Computer Science (London), Techn. Note ICSI 329, 1971. · Zbl 0282.65009
[24] Nicolaides, R.A., On a class of finite elements generated by Lagrange interpolation. To appear. · Zbl 0282.65009
[25] Nitsche, J., Interpolation in Sobolevschen Funktionenräumen. Numer. Math. 13, 334–343 (1969). · Zbl 0221.65012 · doi:10.1007/BF02165408
[26] Nitsche, J., Lineare Spline-Funktionen und die Methode von Ritz für elliptische Randwertprobleme. Arch. Rational Mech. Anal. 36, 348–355 (1970). · Zbl 0192.44503 · doi:10.1007/BF00282271
[27] Salzer, Herbert E., Formulas for bivariate hyperosculatory interpolation. Math. Comp. 25, 119–133 (1971). · Zbl 0252.65003 · doi:10.1090/S0025-5718-1971-0287671-8
[28] Schultz, M. H., L2-multivariate approximation theory. SIAM J. Numer. Anal. 6, 184–209 (1969). · Zbl 0202.15902 · doi:10.1137/0706018
[29] Strang, Gilbert, The finite element method and approximation theory (Symp. on the Numerical Solution of Partial Differential Equations, Univ. of Maryland, 1970). Numerical Solution of Partial Differential Equations-II (B. Hubbard, ed.), p. 547–583. New York: Academic Press 1971.
[30] Strang, Gilbert, & George Fix, Chapter III of: An Analysis of the Finite Element Method. To appear. · Zbl 0272.65099
[31] Taylor, Angus E., Introduction to Functional Analysis. New-York: John Wiley 1958. · Zbl 0081.10202
[32] Thacher, Jr., Henry C., Derivation of interpolation formulas in several independent variables. Ann. New York Acad. Sci. 86, 758–775 (1960). · Zbl 0156.17302
[33] Thacher, Jr., Henry C., & W.E. Milne, Interpolation in several variables. J. Soc. Indust. Appl. Math. 8, 33–42 (1960). · Zbl 0094.31303 · doi:10.1137/0108004
[34] Ženíšek, Alexander, Interpolation polynomials on the triangle. Numer. Math. 15, 283–296 (1970). · Zbl 0216.38901 · doi:10.1007/BF02165119
[35] Zlámal, Miloš, On the finite element method. Numer. Math. 12, 394–409 (1968). · Zbl 0176.16001 · doi:10.1007/BF02161362
[36] Zlámal, Miloš, A finite element procedure of the second order of accuracy. Numer. Math. 14, 394–402 (1970). · Zbl 0209.18002 · doi:10.1007/BF02165594
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